Catalan Numbers Algorithm
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that happen in various counting problems, often involve recursively-specify objects. They are named after the Belgian mathematician Eugène Charles Catalan (1814–1894). The Catalan sequence was described in the 18th century by Leonhard Euler, who was interested in the number of different ways of divide a polygon into triangles. For case, Ming used the Catalan sequence to express series expansions of sin(2α) and sin(4α) in terms of sin(α).In 1988, it get to light that the Catalan number sequence had been used in China by the Mongolian mathematician Mingantu by 1730.
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Finding Nth Catalan number modulo some mod
(N <= 500000, mod is not necessary prime)
Uses Eratosthenes sieve for fast factorization
Works in O(N * loglogN)
Based on problem 140 from acm.mipt.ru
http://acm.mipt.ru/judge/problems.pl?problem=140
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#include <iostream>
#include <fstream>
#include <cmath>
#include <algorithm>
#include <vector>
#include <set>
#include <map>
#include <stack>
#include <queue>
#include <cstdlib>
#include <cstdio>
#include <string>
#include <cstring>
#include <cassert>
#include <utility>
#include <iomanip>
using namespace std;
const int MAXN = 500500;
int n, mod;
int sieve[2 * MAXN];
int p[2 * MAXN];
long long ans = 1;
int main() {
//assert(freopen("input.txt","r",stdin));
//assert(freopen("output.txt","w",stdout));
scanf("%d %d", &n, &mod);
for (int i = 2; i <= 2 * n; i++) {
if (sieve[i] == 0) {
for (int j = 2 * i; j <= 2 * n; j += i) {
if (sieve[j] == 0)
sieve[j] = i;
}
}
}
for (int i = n + 2; i <= 2 * n; i++) {
int x = i;
while (sieve[x] != 0) {
p[sieve[x]]++;
x /= sieve[x];
}
p[x]++;
}
for (int i = 1; i <= n; i++) {
int x = i;
while (sieve[x] != 0) {
p[sieve[x]]--;
x /= sieve[x];
}
p[x]--;
}
for (int i = 2; i <= 2 * n; i++) {
for (int j = 1; j <= p[i]; j++)
ans = (1ll * ans * i) % mod;
}
cout << ans % mod << endl;
return 0;
}
