Catalan(卡特兰)数

定义：令h(1)=1,h(0)=1,Catalan数满足递归式

           h(n)=h(0)*h(n-1)+h(1)*h(n-2)+.....+h(n-1)*h(0) （h≥2）

另类递归式：

          h(n)=((4*n-2)/(n+1))*h(n-1)

该递归关系的解为

         h(n)=C(2n,n)/(n+1)  （n=1,2,3,4.....）

更多关于卡特兰数的资料：

http://blog.youkuaiyun.com/hackbuteer1/article/details/7450250 -Hackbuteer1

http://blog.youkuaiyun.com/ffjjqqjj/article/details/6081711 -ffjjqqjj

题目：http://poj.org/problem?id=2084

      http://acm.hdu.edu.cn/showproblem.php?pid=1023

对前100个卡特兰数打表代码：

#include <iostream>
#include <cstdio>
#include <cstring>

using namespace std;

const int TOP_BIT=100;
const int MOD=10000;
void multi(int a[],int x) //a[]*x
{
	int arry=0;
	for(int i=0;i<TOP_BIT;i++)
	{
		arry+=x*a[i];
		a[i]=arry%MOD;
		arry/=MOD;
	}
}

void divide(int a[],int b) //a[]/b
{
	int divd=0;
	for(int i=TOP_BIT-1;i>=0;i--)
	{
		divd=divd*MOD+a[i];
		a[i]=divd/b;
		divd%=b;
	}
}

int a[100][100];

int main()
{
	
	a[1][0]=1;
	for(int i=2;i<101;i++)
	{
		memcpy(a[i],a[i-1],100*sizeof(int));
		multi(a[i],4*i-2);
		divide(a[i],i+1);
	}
	for(int i=1;i<101;i++)
	{
		int j=99;
		printf("\"");
		while(a[i][j]==0 && j>=0) j--;
		printf("%d",a[i][j]); j--;
		for(;j>=0;j--)
			printf("%04d",a[i][j]);
		printf("\",");
		printf("\n");
	}
	return 0;
}

代码：

#include <iostream>
#include <cstdio>

using namespace std;

char a[][100]={
					"1",
					"2",
					"5",
					"14",
					"42",
					"132",
					"429",
					"1430",
					"4862",
					"16796",
					"58786",
					"208012",
					"742900",
					"2674440",
					"9694845",
					"35357670",
					"129644790",
					"477638700",
					"1767263190",
					"6564120420",
					"24466267020",
					"91482563640",
					"343059613650",
					"1289904147324",
					"4861946401452",
					"18367353072152",
					"69533550916004",
					"263747951750360",
					"1002242216651368",
					"3814986502092304",
					"14544636039226909",
					"55534064877048198",
					"212336130412243110",
					"812944042149730764",
					"3116285494907301262",
					"11959798385860453492",
					"45950804324621742364",
					"176733862787006701400",
					"680425371729975800390",
					"2622127042276492108820",
					"10113918591637898134020",
					"39044429911904443959240",
					"150853479205085351660700",
					"583300119592996693088040",
					"2257117854077248073253720",
					"8740328711533173390046320",
					"33868773757191046886429490",
					"131327898242169365477991900",
					"509552245179617138054608572",
					"1978261657756160653623774456",
					"7684785670514316385230816156",
					"29869166945772625950142417512",
					"116157871455782434250553845880",
					"451959718027953471447609509424",
					"1759414616608818870992479875972",
					"6852456927844873497549658464312",
					"26700952856774851904245220912664",
					"104088460289122304033498318812080",
					"405944995127576985730643443367112",
					"1583850964596120042686772779038896",
					"6182127958584855650487080847216336",
					"24139737743045626825711458546273312",
					"94295850558771979787935384946380125",
					"368479169875816659479009042713546950",
					"1440418573150919668872489894243865350",
					"5632681584560312734993915705849145100",
					"22033725021956517463358552614056949950",
					"86218923998960285726185640663701108500",
					"337485502510215975556783793455058624700",
					"1321422108420282270489942177190229544600",
					"5175569924646105559418940193995065716350",
					"20276890389709399862928998568254641025700",
					"79463489365077377841208237632349268884500",
					"311496878311103321137536291518809134027240",
					"1221395654430378811828760722007962130791020",
					"4790408930363303911328386208394864461024520",
					"18793142726809884575211361279087545193250040",
					"73745243611532458459690151854647329239335600",
					"289450081175264899454283846029490767264392230",
					"1136359577947336271931632877004667456667613940",
					"4462290049988320482463241297506133183499654740",
					"17526585015616776834735140517915655636396234280",
					"68854441132780194707888052034668647142985206100",
					"270557451039395118028642463289168566420671280440",
					"1063353702922273835973036658043476458723103404520",
					"4180080073556524734514695828170907458428751314320",
					"16435314834665426797069144960762886143367590394940",
					"64633260585762914370496637486146181462681535261000",
					"254224158304000796523953440778841647086547372026600",
					"1000134600800354781929399250536541864362461089950800",
					"3935312233584004685417853572763349509774031680023800",
					"15487357822491889407128326963778343232013931127835600",
					"60960876535340415751462563580829648891969728907438000",
					"239993345518077005168915776623476723006280827488229600",
					"944973797977428207852605870454939596837230758234904050",
					"3721443204405954385563870541379246659709506697378694300",
					"14657929356129575437016877846657032761712954950899755100",
					"57743358069601357782187700608042856334020731624756611000",
					"227508830794229349661819540395688853956041682601541047340",
					"896519947090131496687170070074100632420837521538745909320"
	};

int main()
{
	int n,ncase=1,m;
	scanf("%d",&n);
	while(n--)
	{
		scanf("%d",&m);
		printf("Case #%d : %s\n",ncase++,a[m]);
	}
	return 0;
}