【2020模拟赛day7】C. 括号序列

Catalan数其实就这几个常见的应用：

Catn表示长度2n的dyck word的个数。Dyck word是一个有n个X和n个Y组成的字串，且所有的前缀字串皆满足X的个数大于等于Y的个数。以下为长度为6的dyck words:

XXXYYY XYXXYY XYXYXY XXYYXY XXYXYY

将上例的X换成左括号，Y换成右括号，Cn表示所有包含n组括号的合法运算式的个数：

((())) ()(()) ()()() (())() (()())

Catn表示有n个节点组成不同构二叉树（无法通过交换左右儿子互相转换的）的方案数。

Catn表示有2n+1个节点组成不同构满二叉树的方案数。

Catn表示所有在n × n格点中不越过对角线的单调路径的个数。一个单调路径从格点左下角出发，在格点右上角结束，每一步均为向上或向右。计算这种路径的个数等价于计算Dyck word的个数：X代表“向右”，Y代表“向上”。

Catn表示通过连结顶点而将n + 2边的凸多边形分成三角形的方法个数。

三个性质：

$Cat_{n}=\frac{Cat_{n-1}*(4n-2)}{n+1}$

$Cat_{n}=C_{2n}^{n}-C_{2n}^{n+1}$

$Cat_{n}=\frac{C_{2n}^{n}}{n+1}$

然后在简单打个表优化一下就可以了

但是我还是WA了两个点（一会儿改吧）

代码

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int mod=1e9+7;
int ks[4005]={1,993385195,659189998,524916839,18014101,883502100,286641872,954187495,850837389,584887416,335860618,563094647,489743711,521334173,843326065,286617716,749335061,258654345,662907021,707215358,144650061,451452165,378181594,452797764,25110468,163524463,60429994,99238685,267131237,593593917,928117498,527506258,383842542,578251004,822062320,239578401,932375233,855591131,345876444,615334352,318732046,426786847,911684287,316975671,725463896,319038474,640891368,7353552,726441979,626954818,167808926,503390561,869764997,69658095,533073859,498900240,778865557,398018287,765850245,751341869,989944088,667577097,483486099,886251424,970528058,712082734,641209760,115177288,914011496,946337770,236482435,381309250,788928994,204448328,396459081,824441526,384846871,295483431,770392763,611850449,554027257,414802901,773692414,765417640,129962472,488573424,638074197,266756803,626876865,192765465,874984121,944168219,369123925,60481913,148718550,989092825,937873085,125882795,848033029,972225173,222278328,810053131,482472069,565689167,309548182,497980254,346506420,236334053,952251877,40082079,238037566,77878699,110821137,547720725,27325330,733700336,769103266,501254119,785173915,340290324,466803699,252607448,384229381,301391133,367243449,343352642,553839192,719135740,35475352,281911923,580754176,741615984,282630811,268011565,855008763,801038547,513879953,654667491,206775184,170528554,893194234,906513741,322778562,550247701,688090265,217255602,462212929,998742681,181868623,428858037,662442536,531123857,929550716,488038857,804511564,542748977,317167412,917838057,435303256,659191541,419071868,341341470,273963319,282466140,312373327,226436782,627087120,114367572,684357358,251414955,222202249,90440364,913132521,569460926,957350608,46002034,464326151,981274179,512052698,839584048,840409328,5635950,599100549,487012626,595749254,883413861,277832562,440198480,250310206,401217419,886943240,721074837,335026418,503052993,324092758,97775616,80284635,326819157,242106754,235325625,392201000,827710661,538374640,28690354,95437377,100693603,884232876,320668309,76561108,43697508,984010168,117580996,980619098,770971397,991229479,409919529,774014618,39546775,872722605,606553814,862974902,653730165,320149524,931270405,721465406,778715928,362602088,116872526,345105472,100559692,437154486,904552354,935604216,344425405,602977613,599276218,158325398,93069737,433209907,613053094,976515620,60282322,333002046,752010054,379457300,133155766,179241859,709661306,67813055,446653814,78420951,411543902,535315837,117750217,517959892,339342314,385061878,499072682,57377618,968722802,330571243,205863119,783598242,145796116,749295977,526010228,883268039,940907413,344949705,538925361,469031146,953726213,956105552,153838447,913150403,746238927,743089231,362113394,142825047,830251307,583470516,224898926,227622433,351990281,576903763,949020367,772791281,754190225,850436643,164244837,554774537,101010169,52829187,297033729,743020592,715543232,739288258,538040237,108105983,272038670,826616898,623633886,411319195,219758517,773058463,927134472,775457537,340490640,859196509,950335335,168424063,353653970,603402618,224513912,144374618,205611810,787054731,542714651,647021712,330260825,278830139,107669281,270453674,62068059,433486821,839220259,886690937,541379368,99516938,90212094,232966965,928230085,227346281,181140254,719625701,576310761,809002728,769922659,828804701,920385290,767410598,500725173,703550746,504034946,428275786,23085204,869949503,362541383,338787188,204464641,972917583,929397330,653616978,641332697,888596570,696279200,665430171,279314414,198384782,952747077,147269901,933018396,293323438,541567189,685483349,97863149,622160267,448364938,68065202,788470873,757001077,580452048,995437692,504247717,821425587,980313040,404389789,43899747,735342575,838880867,100791819,106424432,805093124,551660012,490088289,612964927,884529047,193556754,465656764,74188019,366369833,584795534,593650941,518920424,712920500,222554786,785561803,596150419,80558548,397073203,840940728,140738130,872279533,46296740,813426497,525107039,157604872,467386603,99859549,413298598,374112248,440635994,290902060,821408912,566460200,847848196,449891426,806911624,913607171,25314236,699849035,47084061,262992150,255863877,635407650,637087249,254294259,856032667,301118483,818265401,49988011,743749904,466376477,362639073,820814395,486190977,454901846,181421642,315020134,380809958,123728381,834494060,196127019,157307939,843056091,113578311,925192595,885708136,846469023,313112308,860735512,621130195,512982422,312344159,565079870,289237615,726458485,77243156,317478856,220084729,404587982,219742437,29992566,534942102,180039049,552648473,834260040,545838564,620572910,362728921,606383070,9870267,698786881,381077417,779866757,506231488,805260820,318345172,9234443,842596294,26429234,541190991,248658537,587810481,865365859,898758874,936437906,956061135,776729249,43506905,684849840,96354867,850853023,133816316,280104305,204611064,974897703,532155072,862278307,809613509,591564705,188373684,419323833,330275712,442417229,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13483452,479826762,730740290,817484630,636300292,424077532,109868049,427612417,851906004,318227793,922677219,672519835,922364580,616438812,88713183,730169238,860094805,856647961,484147888,313300920,354163589,523499291,987276009,830305745,297231672,548606333,906158611,258999573,91087546,311805574,848427919,830083853,487853090,397921540,907527492,772264208,56282335,649517237,588257246,995409529,90944287,781682616,158955222,51049353,369289212,433762399,612634981,719026753,119876110,880892497,671356732,447025697,777893109,343358201,633525895,48831718,667410337,612381345,770290435,979284421,298932395,777810726,510581130,587499198,630150622,296269757,393086393,880397452,874318669,437067210,329351492,323029217,683360968,5271357,754493576,307347243,119498369,253677040,949707339,536075220,469547532,302860254,759274703,728037900,507834940,860405222,933929848,782938240,260014462,352018585,598130613,112221615,171836488,606511552,891430096,41004489,175908097,944018157,804153606,719592370,800248597,726789026,371825258,439990021,248232432,824416165,779344872,135421597,57434699,25085686,989627606,454651478,936864504,981703157,991004666,645620132,954099006,32452938,669733602,154109399,387093216,378995966,342370637,809295425,9309000,115234053,682429037,467050046,143701469,650616409,75050325,297905509,761735656,680319479,655952554,420923228,505021576,566545714,498931165,356270968,10127536,86214567,722110373,637148038,802880664,975324824,929405654,19938269,970586935,831252466,933225061,707947772,704699345,741327190,971660874,511293780,622934128,875021219,640698671,558951109,454121461,281848757,242966096,19676117,404854673,456354781,44441340,190382129,952481058,590890900,951238092,107640928,104300953,939770093,796657151,594411855,423225928,759108006,652556518,334766943,110839958,894789024,642244075,157301604,813191714,184533568,634227997,849667709,618722786,64555561,818082,230838282,161956762,18165083,446260205,113543962,900429131,751473784,764018746,839832535,322771095,909713268,451787926,137449582,301875588,471144891,699447736,903093216,443756052,564891997,144560088,871790474,405216243,259842175,190555880,944827495,940946852,314967381,448335182,757053902,363035327,248321422,173007339,586964885,510197747,699792672,856911686,348181233,317840270,670095799,499295341,69084627,527109696,458470545,512612336,399661310,315930953,708122890,589345486,743187257,219464903,787544726,465653944,87841722,990485901,752248408,715616877,157923640,302789860,366150373,85675378,465330772,463445643,95104439,488059297,217036610,914083959,823624557,989078772,706239813,577838720,979418813,29636711,386563093,383232840,775164344,217293728,327243216,344184436,149296921,867564969,104607624,549059026,832574977,717638293,147458769,692287073,673573351,25365741,653058265,577986622,517386707,187187313,267945481,899352950,585054421,75171736,623799132,180837811,823445226,363817284,116738589,866183428,188996859,827253190,484496812,406659,323182173,158451159,440988449,149150376,507610275,233239888,723990230,318408771,853253535,553267697,131486640,496965176,805910950,613126270,10645395,27053440,579854541,68956800,380112227,545548111,176721485,934723699,20652056,244994610,716623935,56605996,924398368,438572614,345415180,651436872,395229632,607770892,354671742,132277215,642342374,812598882,701183306,384770918,678920854,391516962,460867192,492703176,402180241,365875860,764665366,993985283,12201530,2849200,982241715,380454648,391519276,549323014,951341155,614980327,288220896,598969753,542571039,343910671,324074679,784162949,324629843,118154263,155032769,786470500,576394201,950197330,798544236,121118187,253519960,787886399,424668356,854823206,172549636,150799927,766765801,718396655,183024467,647073713,323707607,134788770,747145728,379923339,481271015,838017427,96668287,310978853,349991207,953826891,630608853,513568896,506387095,742454941,355640128,428410294,727978520,646171336,27386841,647083910,117823016,799775427,294151414,428371897,489650130,141207001,517996566,424423195,988045309,82317638,700468745,907071686,894159539,951475846,664848723,177765306,352348135,472814669,794752132,277421258,452252059,600399156,653868469,776015401,35069889,668744431,176738146,56683333,869510056,958098774,91083111,964360365,227552640,33423545,411059510,392332412,783096015,301115040,120509226,958872584,72070155,759705502,568528406,834043496,161745472,331014619,608505508,872703239,590804404,324634182,167712853,362183309,885524720,974645512,255983822,682229852,489305872,709661040,810267805,271561147,937258824,883298353,367844439,813703360,536322225,291148181,794906902};
void exgcd(ll a,ll b,ll& d,ll& x,ll& y)
{
    if(!b){ d=a; x=1; y=0;}
    else
	{ 
		exgcd(b,a%b,d,y,x); 
		y-=x*(a/b); 
	}
}
ll rev(ll a){
    ll d,x,y;
    exgcd(a,mod,d,x,y);
    return d==1?(x+mod)%mod:-1;
}
int n;
char ss[100005];
int main()
{
	freopen("seq.in","r",stdin);
	freopen("seq.out","w",stdout);
	scanf("%d%s",&n,ss);
	int len=strlen(ss);
	ll ans=(n-len+1)%mod;
	int s=500001;
	ll t=(n-len)/2;
	ll pos=t/s;
	ll resn=ks[pos];
	for(int i=pos*s+1;i<=t;i++)
		resn=(resn*i)%mod;
	t=t*2; pos=t/s;
	ll res2n=ks[pos];
	for(int i=pos*s+1;i<=t;i++)
		res2n=(res2n*i)%mod;
	ll c=res2n;
	c=(c*rev(resn))%mod;
	c=(c+mod)%mod;
	c=(c*rev(resn))%mod;
	c=(c+mod)%mod;
	t/=2;
	ans=(ans*(c*rev(t+1)%mod))%mod;
	printf("%lld",(ans+mod)%mod);
	return 0;	
}

原来是特判不合法的情况题目只保证了S合法！！！

AC代码

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int mod=1e9+7;
int ks[4005]={1,993385195,659189998,524916839,18014101,883502100,286641872,954187495,850837389,584887416,335860618,563094647,489743711,521334173,843326065,286617716,749335061,258654345,662907021,707215358,144650061,451452165,378181594,452797764,25110468,163524463,60429994,99238685,267131237,593593917,928117498,527506258,383842542,578251004,822062320,239578401,932375233,855591131,345876444,615334352,318732046,426786847,911684287,316975671,725463896,319038474,640891368,7353552,726441979,626954818,167808926,503390561,869764997,69658095,533073859,498900240,778865557,398018287,765850245,751341869,989944088,667577097,483486099,886251424,970528058,712082734,641209760,115177288,914011496,946337770,236482435,381309250,788928994,204448328,396459081,824441526,384846871,295483431,770392763,611850449,554027257,414802901,773692414,765417640,129962472,488573424,638074197,266756803,626876865,192765465,874984121,944168219,369123925,60481913,148718550,989092825,937873085,125882795,848033029,972225173,222278328,810053131,482472069,565689167,309548182,497980254,346506420,236334053,952251877,40082079,238037566,77878699,110821137,547720725,27325330,733700336,769103266,501254119,785173915,340290324,466803699,252607448,384229381,301391133,367243449,343352642,553839192,719135740,35475352,281911923,580754176,741615984,282630811,268011565,855008763,801038547,513879953,654667491,206775184,170528554,893194234,906513741,322778562,550247701,688090265,217255602,462212929,998742681,181868623,428858037,662442536,531123857,929550716,488038857,804511564,542748977,317167412,917838057,435303256,659191541,419071868,341341470,273963319,282466140,312373327,226436782,627087120,114367572,684357358,251414955,222202249,90440364,913132521,569460926,957350608,46002034,464326151,981274179,512052698,839584048,840409328,5635950,599100549,487012626,595749254,883413861,277832562,440198480,250310206,401217419,886943240,721074837,335026418,503052993,324092758,97775616,80284635,326819157,242106754,235325625,392201000,827710661,538374640,28690354,95437377,100693603,884232876,320668309,76561108,43697508,984010168,117580996,980619098,770971397,991229479,409919529,774014618,39546775,872722605,606553814,862974902,653730165,320149524,931270405,721465406,778715928,362602088,116872526,345105472,100559692,437154486,904552354,935604216,344425405,602977613,599276218,158325398,93069737,433209907,613053094,976515620,60282322,333002046,752010054,379457300,133155766,179241859,709661306,67813055,446653814,78420951,411543902,535315837,117750217,517959892,339342314,385061878,499072682,57377618,968722802,330571243,205863119,783598242,145796116,749295977,526010228,883268039,940907413,344949705,538925361,469031146,953726213,956105552,153838447,913150403,746238927,743089231,362113394,142825047,830251307,583470516,224898926,227622433,351990281,576903763,949020367,772791281,754190225,850436643,164244837,554774537,101010169,52829187,297033729,743020592,715543232,739288258,538040237,108105983,272038670,826616898,623633886,411319195,219758517,773058463,927134472,775457537,340490640,859196509,950335335,168424063,353653970,603402618,224513912,144374618,205611810,787054731,542714651,647021712,330260825,278830139,107669281,270453674,62068059,433486821,839220259,886690937,541379368,99516938,90212094,232966965,928230085,227346281,181140254,719625701,576310761,809002728,769922659,828804701,920385290,767410598,500725173,703550746,504034946,428275786,23085204,869949503,362541383,338787188,204464641,972917583,929397330,653616978,641332697,888596570,696279200,665430171,279314414,198384782,952747077,147269901,933018396,293323438,541567189,685483349,97863149,622160267,448364938,68065202,788470873,757001077,580452048,995437692,504247717,821425587,980313040,404389789,43899747,735342575,838880867,100791819,106424432,805093124,551660012,490088289,612964927,884529047,193556754,465656764,74188019,366369833,584795534,593650941,518920424,712920500,222554786,785561803,596150419,80558548,397073203,840940728,140738130,872279533,46296740,813426497,525107039,157604872,467386603,99859549,413298598,374112248,440635994,290902060,821408912,566460200,847848196,449891426,806911624,913607171,25314236,699849035,47084061,262992150,255863877,635407650,637087249,254294259,856032667,301118483,818265401,49988011,743749904,466376477,362639073,820814395,486190977,454901846,181421642,315020134,380809958,123728381,834494060,196127019,157307939,843056091,113578311,925192595,885708136,846469023,313112308,860735512,621130195,512982422,312344159,565079870,289237615,726458485,77243156,317478856,220084729,404587982,219742437,29992566,534942102,180039049,552648473,834260040,545838564,620572910,362728921,606383070,9870267,698786881,381077417,779866757,506231488,805260820,318345172,9234443,842596294,26429234,541190991,248658537,587810481,865365859,898758874,936437906,956061135,776729249,43506905,684849840,96354867,850853023,133816316,280104305,204611064,974897703,532155072,862278307,809613509,591564705,188373684,419323833,330275712,442417229,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8,493068714,962438374,640407473,237576280,886732913,104480579,570225220,150635410,825125686,218296091,687815359,836324756,27623309,205369788,497240340,481944296,522665337,214156593,119372468,560466858,509215513,870025939,262338230,544555212,393691698,266026285,959574618,668859633,920800812,833014332,340161002,696424671,595821622,278014196,850282541,465492639,256772637,361536327,407688469,886735505,777677120,559764864,467460478,656780354,296615860,8485023,631513322,892329116,643069866,171347014,301059437,646499816,189218592,157288192,30391941,590039233,127333544,402420074,390773139,279289057,479206985,548838734,58700313,124476754,285841307,832694280,454035278,878625554,195909682,832477285,521190488,616797853,59833194,569019761,985551591,9775115,829731358,367800321,145364434,448786469,583992445,39986993,679085637,26478093,610886157,282629926,721302013,5981636,484998718,304954468,933241589,861237997,306526634,54107029,195156258,703005325,101177873,1600888,328064620,58195526,4016553,672992687,167910811,333963289,651905764,816743397,433830555,988413875,73193416,118874380,871696119,868580380,726537759,50632309,497976091,593811097,57593055,896131555,112246859,286537849,571915555,204577630,819974309,277336537,920087973,599644256,466357030,229839954,716691456,770930126,773937319,488153715,793866053,327909197,302331172,411108160,397100739,579923089,55426222,428689645,469515819,374917698,193588395,810501327,287219964,731387589,668084010,625324906,623993536,195554490,391452013,429389364,733709453,894486530,943000606,435947259,523147733,17645939,317988674,717913197,758380361,134670894,248140767,909109840,354678574,524721605,570756453,61361392,247471050,809826616,421950661,72543708,326944534,691636858,983563134,765698308,985254363,831869335,913276638,62210644,462837721,53408788,167375052,915047828,930108783,817192361,292783094,888348633,772588219,529923072,380565147,389967549,829601921,31868161,259163534,366544644,313671257,661725753,141756944,742432871,189027794,866621856,786299001,938383014,545527799,546815672,913263947,292222814,886085847,360964721,420617382,757531300,150636882,235954771,8750876,344673009,454557914,883413317,828614033,480478115,271504491,9729635,165320936,529426782,966355436,549986905,272262430,461783432,586574579,337961049,935212136,750693581,868837826,275458944,318009087,605837160,536521715,320783373,586322048,336375620,519788282,90881304,809264120,956130992,48076741,980531911,365991811,662281452,963414152,316235883,504253466,124835469,868175104,456045987,611978708,483218123,914255153,504598376,838460049,959815290,654143385,823278276,562973577,513250586,18840615,198694068,60795938,29065702,818615962,390809023,425137900,681467516,24453044,461996349,74666850,729636618,296357442,320390978,873500236,701600816,554867944,82056197,69948524,51025465,249109347,926556414,935118418,881613362,289073770,915958839,655942086,787900869,492468396,342222179,983511014,730767551,717463474,428841574,900312310,228687623,767468415,695956946,351798391,530874744,825919271,244747918,773044091,281912181,780418573,643674314,146940892,621272174,224308302,681262900,598038711,420493769,16054099,808478596,245498105,378224294,618525529,77646971,974239216,889377712,669009052,794893276,851646246,441363191,302675272,129572320,782847588,60821205,347978995,316843448,292441913,279286509,984751679,978150255,803075403,393296319,681058103,550715231,74293983,535461797,242901579,187454961,319283768,493667515,581792964,381242599,572080682,229027458,717348897,675605005,210422310,267915869,539886303,493049383,738868666,340843413,962147401,969857715,530259420,202244520,591146151,994067531,191710738,863433118,543285528,61887548,133284854,701078134,85219070,421849925,545434797,123867200,550611682,319280762,402333062,286576846,149656813,467374693,8399304,675892411,501759115,956075640,652300320,850466095,79872353,755468307,348432666,123146738,23981060,629859587,49085949,85483868,271843323,247494529,968746671,14473670,384928859,947843024,582722209,902859308,550013580,889204720,25094430,804416772,593014077,308719005,376879868,741191647,755350281,274728159,565679798,583556044,139449195,947028131,564955853,92002741,768040234,87026366,819322640,181412802,478920119,567711396,919762601,846135096,185325742,785786737,171574608,356951069,670605801,319513970,857325836,797535716,679276752,453144228,70879217,909853855,731853409,878444861,62980745,222979495,104737293,829984607,878009349,465722368,386425951,861550226,918120693,662828023,487358256,998846726,118976521,311024557,248115301,735716804,10956084,189788526,5621,708910937,28876981,451664374,8214051,398295816,189311069,368625028,229303212,112605837,542573392,348914319,404232226,163560648,90877599,294871478,306028131,343989195,933696701,972838640,551779149,460310637,152121078,637247688,512548093,967711803,488940719,606652566,841857072,97049128,303553213,624885123,229876802,298736911,152806502,919893146,539236843,346933461,673367284,315119462,527988906,609896446,287119402,482590616,581269483,594059810,3061473,813483452,479826762,730740290,817484630,636300292,424077532,109868049,427612417,851906004,318227793,922677219,672519835,922364580,616438812,88713183,730169238,860094805,856647961,484147888,313300920,354163589,523499291,987276009,830305745,297231672,548606333,906158611,258999573,91087546,311805574,848427919,830083853,487853090,397921540,907527492,772264208,56282335,649517237,588257246,995409529,90944287,781682616,158955222,51049353,369289212,433762399,612634981,719026753,119876110,880892497,671356732,447025697,777893109,343358201,633525895,48831718,667410337,612381345,770290435,979284421,298932395,777810726,510581130,587499198,630150622,296269757,393086393,880397452,874318669,437067210,329351492,323029217,683360968,5271357,754493576,307347243,119498369,253677040,949707339,536075220,469547532,302860254,759274703,728037900,507834940,860405222,933929848,782938240,260014462,352018585,598130613,112221615,171836488,606511552,891430096,41004489,175908097,944018157,804153606,719592370,800248597,726789026,371825258,439990021,248232432,824416165,779344872,135421597,57434699,25085686,989627606,454651478,936864504,981703157,991004666,645620132,954099006,32452938,669733602,154109399,387093216,378995966,342370637,809295425,9309000,115234053,682429037,467050046,143701469,650616409,75050325,297905509,761735656,680319479,655952554,420923228,505021576,566545714,498931165,356270968,10127536,86214567,722110373,637148038,802880664,975324824,929405654,19938269,970586935,831252466,933225061,707947772,704699345,741327190,971660874,511293780,622934128,875021219,640698671,558951109,454121461,281848757,242966096,19676117,404854673,456354781,44441340,190382129,952481058,590890900,951238092,107640928,104300953,939770093,796657151,594411855,423225928,759108006,652556518,334766943,110839958,894789024,642244075,157301604,813191714,184533568,634227997,849667709,618722786,64555561,818082,230838282,161956762,18165083,446260205,113543962,900429131,751473784,764018746,839832535,322771095,909713268,451787926,137449582,301875588,471144891,699447736,903093216,443756052,564891997,144560088,871790474,405216243,259842175,190555880,944827495,940946852,314967381,448335182,757053902,363035327,248321422,173007339,586964885,510197747,699792672,856911686,348181233,317840270,670095799,499295341,69084627,527109696,458470545,512612336,399661310,315930953,708122890,589345486,743187257,219464903,787544726,465653944,87841722,990485901,752248408,715616877,157923640,302789860,366150373,85675378,465330772,463445643,95104439,488059297,217036610,914083959,823624557,989078772,706239813,577838720,979418813,29636711,386563093,383232840,775164344,217293728,327243216,344184436,149296921,867564969,104607624,549059026,832574977,717638293,147458769,692287073,673573351,25365741,653058265,577986622,517386707,187187313,267945481,899352950,585054421,75171736,623799132,180837811,823445226,363817284,116738589,866183428,188996859,827253190,484496812,406659,323182173,158451159,440988449,149150376,507610275,233239888,723990230,318408771,853253535,553267697,131486640,496965176,805910950,613126270,10645395,27053440,579854541,68956800,380112227,545548111,176721485,934723699,20652056,244994610,716623935,56605996,924398368,438572614,345415180,651436872,395229632,607770892,354671742,132277215,642342374,812598882,701183306,384770918,678920854,391516962,460867192,492703176,402180241,365875860,764665366,993985283,12201530,2849200,982241715,380454648,391519276,549323014,951341155,614980327,288220896,598969753,542571039,343910671,324074679,784162949,324629843,118154263,155032769,786470500,576394201,950197330,798544236,121118187,253519960,787886399,424668356,854823206,172549636,150799927,766765801,718396655,183024467,647073713,323707607,134788770,747145728,379923339,481271015,838017427,96668287,310978853,349991207,953826891,630608853,513568896,506387095,742454941,355640128,428410294,727978520,646171336,27386841,647083910,117823016,799775427,294151414,428371897,489650130,141207001,517996566,424423195,988045309,82317638,700468745,907071686,894159539,951475846,664848723,177765306,352348135,472814669,794752132,277421258,452252059,600399156,653868469,776015401,35069889,668744431,176738146,56683333,869510056,958098774,91083111,964360365,227552640,33423545,411059510,392332412,783096015,301115040,120509226,958872584,72070155,759705502,568528406,834043496,161745472,331014619,608505508,872703239,590804404,324634182,167712853,362183309,885524720,974645512,255983822,682229852,489305872,709661040,810267805,271561147,937258824,883298353,367844439,813703360,536322225,291148181,794906902};
void exgcd(ll a,ll b,ll& d,ll& x,ll& y)
{
    if(!b){ d=a; x=1; y=0;}
    else
	{ 
		exgcd(b,a%b,d,y,x); 
		y-=x*(a/b); 
	}
}
ll rev(ll a){
    ll d,x,y;
    exgcd(a,mod,d,x,y);
    return d==1?(x+mod)%mod:-1;
}
int n;
char ss[100005];
int main()
{
	freopen("seq.in","r",stdin);
	freopen("seq.out","w",stdout);
	scanf("%d%s",&n,ss);
	int len=strlen(ss);
	if((n&1)||n<len)
	{
		puts("0"); 
		return 0; 
	}
	ll ans=(n-len+1)%mod;
	int s=500001;
	ll t=(n-len)/2;
	ll pos=t/s;
	ll resn=ks[pos];
	for(int i=pos*s+1;i<=t;i++)
		resn=(resn*i)%mod;
	t=t*2; pos=t/s;
	ll res2n=ks[pos];
	for(int i=pos*s+1;i<=t;i++)
		res2n=(res2n*i)%mod;
	ll c=res2n;
	c=(c*rev(resn))%mod;
	c=(c+mod)%mod;
	c=(c*rev(resn))%mod;
	c=(c+mod)%mod;
	t/=2;
	ans=(ans*(c*rev(t+1)%mod))%mod;
	printf("%lld",(ans+mod)%mod);
	return 0;	
}

【2020模拟赛day7】C. 括号序列

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