Problem Description
对于一个串S，当它同时满足如下条件时，它就是一个01偏串： 1、只由0和1两种符组成； 2、在S的每一个前缀中，0的个数不超过1的个数； 3、S中0的个数和1的个数相等。
现在给定01偏串S，请计算一下S在所有长度为n的01偏串中作为子串出现的次数的总和。 由于结果比较大，结果对1e9+7取余后输出。
样例解释： 在第二个样例中，长度为4的偏串共两个1010，1100。10在1010中出现了两次，在1100中出现了1次。所以答案是3。
Input
第一行给出一个整数T(1<=T<=40),表示测试数据的数目。 每一组测试包含一个整数n和字符串S，中间用空格分开。(1<=|S|<=100000，1<=n<=1000000000)
输入保证S是一个01偏串。
Output
对于每一组数据，输出一个整数占一行，表示答案。
Sample Input
2 2 10 4 10
Sample Output
1 3解法：1 我们求等于0的情况 n为奇数或者字符长度>n2 我们暴力的算一算发现，这个和字符串长度有关，和具体内容无关3 通过查询发现这是个ans=(n-len)/2
但数据太大，1s算不完，只能化简看看啦4 x=(n-len)/2+1有
我们有个叫卡特兰数的东西
带进去
5 我们只需要算f(x),他有一个
但从1算到1e9还是大了，我们就分段计算，每1e5个数字保存一次，然后每一次计算找到它最近的起点就行
1 #include<bits/stdc++.h>
2 using namespace std;
3 typedef long long ll;
4 //计算n的k次方对M取模，二分法
5 ll Pow(ll n, ll k, ll p){
6 ll ans=1;
7 while(k){
8 if(k&1)
9 ans=(ans*n)%p;
10 n = (n*n)%p;
11 k>>=1; //k=k>>1 k=k/2；
12 }
13 return ans;
14 }
15 ll C(ll n,ll m,ll p){
16 if(m==0) return 1;
17 if(m>n-m) m=n-m;
18 ll up=1,down=1;
19 for(int i=1;i<=m;i++){
20 up = (up*(n-i+1))%p;
21 down=(down*i)%p;
22 }
23 return up*Pow(down,p-2,p)%p;
24 }
25 ll lucas(ll n,ll m,ll p){
26 if(m==0) return 1;
27 return C(n%p,m%p,p)*lucas(n/p,m/p,p);
28 }
29 #define mod 1000000007
30 int ct[6000]={1,945729344,874271714,728937729,488957151,213941567,621948608,165410919,130152966,477849061,70646122,104331417,804334677,112228257,406385851,373610584,959189339,129232103,596011761,653302265,502350404,254254274,23933464,200023312,437339905,838455202,353529507,181826790,268707842,918554769,361346624,475114772,551764204,264122255,348587014,339775734,190610810,147039338,576212642,958452888,907080272,897468755,485757524,926001334,253159604,720502565,795701677,485379561,512490195,62869610,44647373,170729326,181183678,223602264,14266485,546070015,669748641,266401585,722432477,844276843,806809752,215407850,514089950,48258832,388402233,287386547,184275733,39087220,221157750,796058916,274451201,551481903,468840632,654694122,349643823,123450237,466554578,613371117,104229634,534520804,340376885,53768135,665006362,362834988,113383960,458629513,877935697,990187981,24010351,477599350,491321663,619368665,730486899,302800222,248364522,617385859,136706557,988949479,681384128,140672609,39443745,327846413,353172956,108275948,378351259,482275913,432213793,904945283,871063203,757590126,541675614,649654378,339103071,153511190,307061975,535998427,953278988,315396128,629843192,166102135,623766767,9922024,962639966,258645380,484156059,75913669,837794293,574075634,404396328,768092500,903633183,512519672,44285819,538902069,621789723,71341556,81438997,28847533,40513591,849551327,716939958,942552133,395636757,409774346,383255288,612104482,95851638,696370843,884779483,524922348,220269701,390881347,114227221,395767320,614338019,380172060,180822541,691300228,891829295,967678016,701386253,162940737,994175698,908221589,19098271,438672588,631720803,573850469,339056525,473356393,859071294,949727021,98529638,168890563,563975847,7483795,276476730,39489314,514728211,87529508,963753157,380031470,856696972,158703490,515908206,227044626,959511111,993078622,80360936,888779384,323285665,543535546,550763954,601069950,458756853,717452438,141044757,197501322,634124805,238583278,200414323,553793080,66504653,42069842,722598090,913608032,740683059,742050754,961341417,612741304,91785690,526087340,214299335,290988315,920349511,650032143,826861662,655676777,832872230,243503280,428075391,780176613,762279490,958553222,298583588,683539087,245173409,561149407,172900450,994863083,373712992,577710034,676813275,65061172,354158163,474587452,779048700,252731521,513247255,63716789,589280106,300449755,843219077,437936157,617131692,778975010,166446598,341293627,939851882,377019792,974135048,18159888,617899054,824539871,922256819,904951680,298436189,877366713,763013330,69488053,73161945,968841940,502298660,815060335,585478837,52225572,617841877,964952304,864682249,537138023,438032312,507344261,611203109,211965075,510592131,253042462,662211654,306668985,993054101,886017682,550897294,748379779,600811784,662510825,923914254,398819263,459843924,563578785,800110936,890507874,833696238,67588077,662523469,355339772,897895240,326276953,66185485,707119691,543399352,359252814,343020869,435649016,326479513,313711009,180736295,933688638,341979700,832484396,852821336,938966456,327938691,897898594,514258765,67631305,832209634,147167201,933864871,764372915,290610995,894446445,552066969,955550128,899538317,98587581,791520663,371485442,250166899,386277652,850700239,24883422,365069578,542759096,393648297,338311897,998392022,963038213,184331347,216025562,616142588,243671535,922469047,642896156,173840546,617065897,502916270,399859525,626708892,203438585,794335044,466847771,485435744,176762177,370414892,387922877,421040690,506803021,831791096,536761402,659624740,205742555,354512880,992080290,655638247,441557017,45234385,602453687,642559130,907907516,150309994,959349261,667245533,336984599,7514180,293021893,527065808,716061908,183191397,64976751,577967909,833760679,814950379,238991246,714290102,902231891,437427307,149073027,103173270,685925477,694198173,725256565,71168984,261760780,55757530,112359206,436686978,516723667,537954237,916255072,34043384,456908325,881291767,146092388,189805926,498850664,471576564,977024064,502710283,230937450,166248501,100683033,725464484,750575586,525293805,107750423,541103694,100780834,184498203,767490572,317419646,113029816,182757677,138614676,694365799,832641460,450632411,301670090,972578606,809851941,356139480,741575534,319109663,491688831,946837771,820876500,944731056,441691410,439779726,878100434,110259934,781810231,822737100,122949140,185157201,147977057,78103215,98159640,570747872,913214189,526507036,614910594,606219102,234699129,882201646,184318349,623751699,220799331,756153289,390599249,416019375,679472812,608212465,946818526,676784388,848462543,701625645,631586665,279738981,507735946,983863013,797150469,667058957,302361878,11123699,265956565,759232458,630434107,603234456,651468455,184824418,604726700,101186449,24954474,873710205,172883575,299713548,965606121,589841373,782619807,545740664,94087806,846145323,99057958,275906334,98324641,712417499,506415720,153372884,988856645,995466415,137196894,296743168,461577383,447394859,350928543,800328411,35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1,840729498,935684923,155577573,734434833,191517923,654123163,637896624,852058670,173168370,730184794,50915735,324841114,628137110,205149114,182695122,548334792,230832142,246897375,673812822,774458235,12653594,589139188,823592104,141584849,661263989,634220976,838048410,865644208,959538649,496526723,706515642,30364288,288996675,618298630,512675413,255909709,961226649,196140093,464842799,73771491,907822468,820941783,633421126,938423057,725611016,889132555,244245553,600535214,454683334,918501588,156397960,727549083,462344797,552581171,951585647,331017637,713844297,201014346,220893764,942832265,205089583,855055766,558242708,467800323,544040124,5018684,600324086,275710404,942082597,882387855,283394592,984698042,699687145,218659340,671476043,786242938,615660310,451633451,264041371,62057607,615318684,358229710,600528374,737949511,309417464,928413938,485704507,508289212,335423875,674380863,518017044,336119074,617416674,485819110,941096637,929311054,597002446,973475334,265580872,563517784,251569149,148530184,95396816,883600979,282561244,840871566,519908445,640741041,421822961,632376045,139437876,790411904,591820303,741935921,948651905,366299783,716525381,15431695,15465625,971346451,168728596,608506519,27394837,851151360,602776233,970664278,322046939,511916124,340275737,1956214,363111458,481072216,658163046,518552634,132664065,981823198,216243081,159795020,767941105,457986980,858463537,976190745,171187323,572871813,647899578,408338249,44147207,156374207,940179853,58043446,28385650,424517633,511617049,926045094,319879729,514748429,614317869,421026423,671332980,6835612,938930076,458303594,925057217,380172144,417162197,8078854,424003560,2008065,807569105,429700641,464740805,457368858,589073747,394260986,391506131,929826825,432872648,690774587,357406694,692130031,273616783,445027156,511488576,31976171,170358186,20482390,918198464,271438684,749710986,389073776,939445726,23455846,956187346,748137028,734748051,303817400,94204386,126522637,81942995,832809073,197225686,406203721,149560387,711052519,501674175,547993890,871374386,551852029,842913905,11301805,658024191,322045818,938590456,235068927,986201888,729075821,230293940,88489051,430417352,457640653,512463037,164939381,78855727,918299124,19019884,318003658,964003586,887976673,465062825,998287188,345218097,418436509,423868376,737852682,222443100,431664080,191790675,179137865,240191988,426955600,372855698,834981397,769820197,12457840,592849249,301573725,529831284,455945822,908361093,348036200,725019234,272002918,302014486,868309988,267131588,110362906,658933926,714755636,575129133,572255614,242451399,460854477,379734453,261883330,173252303,205271891,875823576,295295415,344691305,88281196,291229175,339458347,199662270,135955543,945346312,464533492,871840955,247580453,899980261,309580183,86978522,845735069,219736536,817617603,405963484,880265007,291366034,314006019,86753505,849126155,994666022,724217006,569965800,190010599,920028225,755930223,376881817,18255664,401402435,819561952,810240409,896675374,5648890,14322856,899043492,141273287,496383310,791004873,659659128,952106154,219105831,176538445,879815182,445593480,150528189,94493662,797589952,732282876,567925839,796782054,629643845,130694449,475573885,497611040,499200426,183468775,892394136,17028961,825354772,382036602,10367014,453970224,702396666,603040526,616353655,902691380,983386965,692566149,247825935,66330622,225112465,895795849,52320773,155947313,718224149,164300545,677709978,266340441,218514417,204476400,286595553,643512303,598766962,788301683,892598856,520036990,859133788,979526775,296869156,670015326,164319827,647208607,679949247,172064170,865163183,337039103,105046080,579964945,123270759,385113924,576031432,771688525,704910535,414466742,609386035,928631412,485368568,479480031,505949447,116422341,568991478,93574014,961174363,735174682,235632013,19234330,650946685,144392601,510837538,567469814,856042956,806198989,116886696,207340279,223268817,253945420,584643811,616483622,711150993,357748602,626063923,607222114,600586776,28542569,796083795,549421818,591429425,607254162,355391295,492527448,847958740,287070353,814226472,643144229,698819990,177700395,611101671,490661583,275559707,950224920,844587825,53229930,339523379,979109376,126378471,444745387,726629825,351823753,475484131,518938465,963071536,84665553,643867081,825793617,408943841,879707301,471828659,223598556,384354814,153231894,188992442,483627866,255018189,675911377,500327547,638084035,486199656,903986764,949245648,862319148,790947898,142578126};
31 ll Pow(ll a, ll b)
32 {
33 ll sum = 1;
34 while(b)
35 {
36 if(b%2)
37 sum = sum*a%mod;
38 a = a*a%mod;
39 b /= 2;
40 }
41 return sum;
42 }
43 int main(){
44 int t=1;
45 char s[123456];
46 ll m,n,ans;
47 while(~scanf("%d",&t)){
48 while(t--){
49 scanf("%lld%s",&n,s);
50 m=strlen(s);
51 //cout<<m<<endl;
52 if(m>n||n&1){
53 printf("0\n");
54 }else{
55 ans=(n-m)/2;
56 if(ans<=4999999){
57 printf("%lld\n",lucas(2*ans+1,ans,1e9+7));
58 }else{
59 int i=1;
60 ans+=1;
61 for(i=1;i<=5000;i++){
62 if(i*100000>ans){
63 break;
64 }
65 }
66 i-=1;
67 // cout<<i<<endl;
68 ll now=ct[i];
69 for(int j=i*100000+1;j<=ans;j++){
70 // cout<<"B"<<endl;
71 now = 2*(2*j-1)*now%mod*Pow(j+1, mod-2)%mod;
72 }
73 ll x=now%mod*(((ans+1)%mod*500000004)%mod)%mod;
74 cout<<x<<endl;
75 }
76 }
77 }
78 }
79 return 0;
80 }
时间： 2024-10-07 21:06:12