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。
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
Source
思路：由偏串的定义知长为2 * l的偏串个数即为卡特兰数第l项，先将原串n - len的部分选好，然后将当前串插入到n - len + 1个空里。不用担心这样得到的串会有重复，因为我们求的不是原串的种类数，而是子串在原串的出现次数，这样得到的结果刚好与所求的一一对应。卡特兰数第h项为C(2 * h, h) / (h + 1)，经化简最终答案为C(2 * h + 1， h)(h = (n - len) / 2)，因为可能求到1e9的阶乘，故需分段打表。
#include <iostream>
#include <fstream>
#include <sstream>
#include <cstdlib>
#include <cstdio>
#include <cmath>
#include <string>
#include <cstring>
#include <algorithm>
#include <queue>
#include <stack>
#include <vector>
#include <set>
#include <map>
#include <iomanip>
#include <cctype>
#include <cassert>
#include <bitset>
#include <ctime>
using namespace std;
#define pau system("pause")
#define ll long long
#define pii pair<int, int>
#define pb push_back
#define mp make_pair
#define clr(a, x) memset(a, x, sizeof(a))
const double pi = acos(-1.0);
const int INF = 0x3f3f3f3f;
const int MOD = 1e9 + 7;
const double EPS = 1e-9;
int t, n, l;
char s[100015];
int res[1015] = {
1,
641102369, 578095319, 5832229, 259081142, 974067448, 316220877, 690120224, 251368199, 980250487, 682498929,
134623568, 95936601, 933097914, 167332441, 598816162, 336060741, 248744620, 626497524, 288843364, 491101308,
245341950, 565768255, 246899319, 968999, 586350670, 638587686, 881746146, 19426633, 850500036, 76479948,
268124147, 842267748, 886294336, 485348706, 463847391, 544075857, 898187927, 798967520, 82926604, 723816384,
156530778, 721996174, 299085602, 323604647, 172827403, 398699886, 530389102, 294587621, 813805606, 67347853,
497478507, 196447201, 722054885, 228338256, 407719831, 762479457, 746536789, 811667359, 778773518, 27368307,
438371670, 59469516, 5974669, 766196482, 606322308, 86609485, 889750731, 340941507, 371263376, 625544428,
788878910, 808412394, 996952918, 585237443, 1669644, 361786913, 480748381, 595143852, 837229828, 199888908,
526807168, 579691190, 145404005, 459188207, 534491822, 439729802, 840398449, 899297830, 235861787, 888050723,
656116726, 736550105, 440902696, 85990869, 884343068, 56305184, 973478770, 168891766, 804805577, 927880474,
876297919, 934814019, 676405347, 567277637, 112249297, 44930135, 39417871, 47401357, 108819476, 281863274,
60168088, 692636218, 432775082, 14235602, 770511792, 400295761, 697066277, 421835306, 220108638, 661224977,
261799937, 168203998, 802214249, 544064410, 935080803, 583967898, 211768084, 751231582, 972424306, 623534362,
335160196, 243276029, 554749550, 60050552, 797848181, 395891998, 172428290, 159554990, 887420150, 970055531,
250388809, 487998999, 856259313, 82104855, 232253360, 513365505, 244109365, 1559745, 695345956, 261384175,
849009131, 323214113, 747664143, 444090941, 659224434, 80729842, 570033864, 664989237, 827348878, 195888993,
576798521, 457882808, 731551699, 212938473, 509096183, 827544702, 678320208, 677711203, 289752035, 66404266,
555972231, 195290384, 97136305, 349551356, 785113347, 83489485, 66247239, 52167191, 307390891, 547665832,
143066173, 350016754, 917404120, 296269301, 996122673, 23015220, 602139210, 748566338, 187348575, 109838563,
574053420, 105574531, 304173654, 542432219, 34538816, 325636655, 437843114, 630621321, 26853683, 933245637,
616368450, 238971581, 511371690, 557301633, 911398531, 848952161, 958992544, 925152039, 914456118, 724691727,
636817583, 238087006, 946237212, 910291942, 114985663, 492237273, 450387329, 834860913, 763017204, 368925948,
475812562, 740594930, 45060610, 806047532, 464456846, 172115341, 75307702, 116261993, 562519302, 268838846,
173784895, 243624360, 61570384, 481661251, 938269070, 95182730, 91068149, 115435332, 495022305, 136026497,
506496856, 710729672, 113570024, 366384665, 564758715, 270239666, 277118392, 79874094, 702807165, 112390913,
730341625, 103056890, 677948390, 339464594, 167240465, 108312174, 839079953, 479334442, 271788964, 135498044,
277717575, 591048681, 811637561, 353339603, 889410460, 839849206, 192345193, 736265527, 316439118, 217544623,
788132977, 618898635, 183011467, 380858207, 996097969, 898554793, 335353644, 54062950, 611251733, 419363534,
965429853, 160398980, 151319402, 990918946, 607730875, 450718279, 173539388, 648991369, 970937898, 500780548,
780122909, 39052406, 276894233, 460373282, 651081062, 461415770, 358700839, 643638805, 560006119, 668123525,
686692315, 673464765, 957633609, 199866123, 563432246, 841799766, 385330357, 504962686, 954061253, 128487469,
685707545, 299172297, 717975101, 577786541, 318951960, 773206631, 306832604, 204355779, 573592106, 30977140,
450398100, 363172638, 258379324, 472935553, 93940075, 587220627, 776264326, 793270300, 291733496, 522049725,
579995261, 335416359, 142946099, 472012302, 559947225, 332139472, 499377092, 464599136, 164752359, 309058615,
86117128, 580204973, 563781682, 954840109, 624577416, 895609896, 888287558, 836813268, 926036911, 386027524,
184419613, 724205533, 403351886, 715247054, 716986954, 830567832, 383388563, 68409439, 6734065, 189239124,
68322490, 943653305, 405755338, 811056092, 179518046, 825132993, 343807435, 985084650, 868553027, 148528617,
160684257, 882148737, 591915968, 701445829, 529726489, 302177126, 974886682, 241107368, 798830099, 940567523,
11633075, 325334066, 346091869, 115312728, 473718967, 218129285, 878471898, 180002392, 699739374, 917084264,
856859395, 435327356, 808651347, 421623838, 105419548, 59883031, 322487421, 79716267, 715317963, 429277690,
398078032, 316486674, 384843585, 940338439, 937409008, 940524812, 947549662, 833550543, 593524514, 996164327,
987314628, 697611981, 636177449, 274192146, 418537348, 925347821, 952831975, 893732627, 1277567, 358655417,
141866945, 581830879, 987597705, 347046911, 775305697, 125354499, 951540811, 247662371, 343043237, 568392357,
997474832, 209244402, 380480118, 149586983, 392838702, 309134554, 990779998, 263053337, 325362513, 780072518,
551028176, 990826116, 989944961, 155569943, 596737944, 711553356, 268844715, 451373308, 379404150, 462639908,
961812918, 654611901, 382776490, 41815820, 843321396, 675258797, 845583555, 934281721, 741114145, 275105629,
666247477, 325912072, 526131620, 252551589, 432030917, 554917439, 818036959, 754363835, 795190182, 909210595,
278704903, 719566487, 628514947, 424989675, 321685608, 50590510, 832069712, 198768464, 702004730, 99199382,
707469729, 747407118, 302020341, 497196934, 5003231, 726997875, 382617671, 296229203, 183888367, 703397904,
552133875, 732868367, 350095207, 26031303, 863250534, 216665960, 561745549, 352946234, 784139777, 733333339,
503105966, 459878625, 803187381, 16634739, 180898306, 68718097, 985594252, 404206040, 749724532, 97830135,
611751357, 31131935, 662741752, 864326453, 864869025, 167831173, 559214642, 718498895, 91352335, 608823837,
473379392, 385388084, 152267158, 681756977, 46819124, 313132653, 56547945, 442795120, 796616594, 256141983,
152028387, 636578562, 385377759, 553033642, 491415383, 919273670, 996049638, 326686486, 160150665, 141827977,
540818053, 693305776, 593938674, 186576440, 688809790, 565456578, 749296077, 519397500, 551096742, 696628828,
775025061, 370732451, 164246193, 915265013, 457469634, 923043932, 912368644, 777901604, 464118005, 637939935,
956856710, 490676632, 453019482, 462528877, 502297454, 798895521, 100498586, 699767918, 849974789, 811575797,
438952959, 606870929, 907720182, 179111720, 48053248, 508038818, 811944661, 752550134, 401382061, 848924691,
764368449, 34629406, 529840945, 435904287, 26011548, 208184231, 446477394, 206330671, 366033520, 131772368,
185646898, 648711554, 472759660, 523696723, 271198437, 25058942, 859369491, 817928963, 330711333, 724464507,
437605233, 701453022, 626663115, 281230685, 510650790, 596949867, 295726547, 303076380, 465070856, 272814771,
538771609, 48824684, 951279549, 939889684, 564188856, 48527183, 201307702, 484458461, 861754542, 326159309,
181594759, 668422905, 286273596, 965656187, 44135644, 359960756, 936229527, 407934361, 267193060, 456152084,
459116722, 124804049, 262322489, 920251227, 816929577, 483924582, 151834896, 167087470, 490222511, 903466878,
361583925, 368114731, 339383292, 388728584, 218107212, 249153339, 909458706, 322908524, 202649964, 92255682,
573074791, 15570863, 94331513, 744158074, 196345098, 334326205, 9416035, 98349682, 882121662, 769795511,
231988936, 888146074, 137603545, 582627184, 407518072, 919419361, 909433461, 986708498, 310317874, 373745190,
263645931, 256853930, 876379959, 702823274, 147050765, 308186532, 175504139, 180350107, 797736554, 606241871,
384547635, 273712630, 586444655, 682189174, 666493603, 946867127, 819114541, 502371023, 261970285, 825871994,
126925175, 701506133, 314738056, 341779962, 561011609, 815463367, 46765164, 49187570, 188054995, 957939114,
64814326, 933376898, 329837066, 338121343, 765215899, 869630152, 978119194, 632627667, 975266085, 435887178,
282092463, 129621197, 758245605, 827722926, 201339230, 918513230, 322096036, 547838438, 985546115, 852304035,
593090119, 689189630, 555842733, 567033437, 469928208, 212842957, 117842065, 404149413, 155133422, 663307737,
208761293, 206282795, 717946122, 488906585, 414236650, 280700600, 962670136, 534279149, 214569244, 375297772,
811053196, 922377372, 289594327, 219932130, 211487466, 701050258, 398782410, 863002719, 27236531, 217598709,
375472836, 810551911, 178598958, 247844667, 676526196, 812283640, 863066876, 857241854, 113917835, 624148346,
726089763, 564827277, 826300950, 478982047, 439411911, 454039189, 633292726, 48562889, 802100365, 671734977,
945204804, 508831870, 398781902, 897162044, 644050694, 892168027, 828883117, 277714559, 713448377, 624500515,
590098114, 808691930, 514359662, 895205045, 715264908, 628829100, 484492064, 919717789, 513196123, 748510389,
403652653, 574455974, 77123823, 172096141, 819801784, 581418893, 15655126, 15391652, 875641535, 203191898,
264582598, 880691101, 907800444, 986598821, 340030191, 264688936, 369832433, 785804644, 842065079, 423951674,
663560047, 696623384, 496709826, 161960209, 331910086, 541120825, 951524114, 841656666, 162683802, 629786193,
190395535, 269571439, 832671304, 76770272, 341080135, 421943723, 494210290, 751040886, 317076664, 672850561,
72482816, 493689107, 135625240, 100228913, 684748812, 639655136, 906233141, 929893103, 277813439, 814362881,
562608724, 406024012, 885537778, 10065330, 60625018, 983737173, 60517502, 551060742, 804930491, 823845496,
727416538, 946421040, 678171399, 842203531, 175638827, 894247956, 538609927, 885362182, 946464959, 116667533,
749816133, 241427979, 871117927, 281804989, 163928347, 563796647, 640266394, 774625892, 59342705, 256473217,
674115061, 918860977, 322633051, 753513874, 393556719, 304644842, 767372800, 161362528, 754787150, 627655552,
677395736, 799289297, 846650652, 816701166, 687265514, 787113234, 358757251, 701220427, 607715125, 245795606,
600624983, 10475577, 728620948, 759404319, 36292292, 491466901, 22556579, 114495791, 647630109, 586445753,
482254337, 718623833, 763514207, 66547751, 953634340, 351472920, 308474522, 494166907, 634359666, 172114298,
865440961, 364380585, 921648059, 965683742, 260466949, 117483873, 962540888, 237120480, 620531822, 193781724,
213092254, 107141741, 602742426, 793307102, 756154604, 236455213, 362928234, 14162538, 753042874, 778983779,
25977209, 49389215, 698308420, 859637374, 49031023, 713258160, 737331920, 923333660, 804861409, 83868974,
682873215, 217298111, 883278906, 176966527, 954913, 105359006, 390019735, 10430738, 706334445, 315103615,
567473423, 708233401, 48160594, 946149627, 346966053, 281329488, 462880311, 31503476, 185438078, 965785236,
992656683, 916291845, 881482632, 899946391, 321900901, 512634493, 303338827, 121000338, 967284733, 492741665,
152233223, 165393390, 680128316, 917041303, 532702135, 741626808, 496442755, 536841269, 131384366, 377329025,
301196854, 859917803, 676511002, 373451745, 847645126, 823495900, 576368335, 73146164, 954958912, 847549272,
241289571, 646654592, 216046746, 205951465, 3258987, 780882948, 822439091, 598245292, 869544707, 698611116
};
ll mpow(ll x, int y) {
if (!y) return 1;
ll res = mpow(x, y >> 1);
if (y & 1) return res * res % MOD * x % MOD;
else return res * res % MOD;
}
ll get_val(int x) {
int y = x / 1000000;
ll result = res[y];
for (int i = y * 1000000 + 1; i <= x; ++i) {
result = result * i % MOD;
}
return result;
}
int main() {
scanf("%d", &t);
while (t--) {
scanf("%d%s", &n, s + 1);
l = strlen(s + 1);
if (l > n || n & 1) {
puts("0");
continue;
} else {
int h = n - l >> 1;
if ((h << 1 | 1) >= MOD) {
puts("0");
continue;
} else {
ll x = get_val(h << 1 | 1);
ll y = get_val(h + 1);
ll z = get_val(h);
ll answer = x * mpow(y, MOD - 2) % MOD * mpow(z, MOD - 2) % MOD;
printf("%I64d
", answer);
}
}
}
return 0;
}