1.那罗延数N(n,k)的计算公式:
N(n,k) = 1/n * C(n,k) * C(n,k-1)
2.那罗延三角前八项:
n\k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
1 | 1 | |||||||
2 | 1 | 1 | ||||||
3 | 1 | 3 | 1 | |||||
4 | 1 | 6 | 6 | 1 | ||||
5 | 1 | 10 | 20 | 10 | 1 | |||
6 | 1 | 15 | 50 | 50 | 15 | 1 | ||
7 | 1 | 21 | 105 | 175 | 105 | 21 | 1 | |
8 | 1 | 28 | 196 | 490 | 490 | 196 | 28 | 1 |
3.应用:
在由n对"("和")"组成的字符串中,共有k对"("与")"相邻,这样的字符串一共有N(n,k)个。例如n=4,k=2时,N(n,k)=6
4.性质:
那罗延三角中每一行的和为卡特兰数,即
N(n,1) + N(n,2) + N(n,3) + ... + N(n,n) = Catalan(n)