A113129 Triangle T(n,k), 0<=k<=n, of coefficients of polynomials P_n(x) related to convolution of the k-fold factorials.

1, 0, 1, 0, 0, 2, 0, 0, 1, 6, 0, 0, 0, 10, 24, 0, 0, 0, 4, 82, 120, 0, 0, 0, 0, 84, 672, 720, 0, 0, 0, 0, 27, 1236, 5820, 5040, 0, 0, 0, 0, 0, 930, 16328, 54288, 40320, 0, 0, 0, 0, 0, 248, 20850, 211080, 548496, 362880, 0, 0, 0, 0, 0, 0, 12452, 396528, 2775432

Offset: 0

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

• easy
• nonn
• tabl

Comments

Let R(m,n,k), 0<=k<=n, the Riordan array (1,x*g(x)) where g(x) is g.f. of the m-fold factorials . Then R(m,n,k) = R(m,n-1,k-1) + Sum_{j, 0<=j<=n-1-k} R(m,n-1,k+j)*P_m(j), R(m,n,0) = 0^n and R(m,0,k) = 0 if k>n.

Examples

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 6;
.0, 0, 0, 10, 24;
.0, 0, 0, 4, 82, 120;
.0, 0, 0, 0, 84, 672, 720;
.0, 0, 0, 0, 27, 1236, 5820, 5040;
.0, 0, 0, 0, 0, 930, 16328, 54288, 40320;
.0, 0, 0, 0, 0, 248, 20850, 211080, 548496, 362880;
.0, 0, 0, 0, 0, 0, 12452, 396528, 2775432, 6003360, 362880;
.0, 0, 0, 0, 0, 0, 2830, 38732, 7057308, 37831752, 71019360, 39916800;
		

Crossrefs

Cf. A000007, A000108, A000142, A000699.

R(m, n, k) : A097805 (m=0), A084938 (m=1), A111106 (m=2), A113333 (column sums).

Formula

P_0(x) = 1, P_1(x) = x, P_2(x) = 2*x^2, P_ n(x) = n*x*P_(n-1)(x) + Sum_{j, 1<=j<=n-1} j*P_j(x)*P_(n-1-j)(x).

P_n(x) = Sum_{k, 0<=k<=n} T(n, k)*x^k.

P_n(0) = A000007(n).

P_n(x) = A075834(n+1), A111088(n+1), A113130(n+1), A113131(n+1), A113132(n+1), A113133(n+1), A113134(n+1), A113135(n+1) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.

P_n(-1) = (-1)^n*A000108(n), signed Catalan numbers.

T(n, n) = n! = A000142(n).

T(2*n+1, n+1) = A000699(n+1) (number of irreducible diagrams with 2n+2 nodes).

T(2*n+2, n+2) = A113332(n) = A000699(n+2)*(2*n+3)*(n+2)/(3*(n+1)).

Extensions

Corrected by Philippe Deléham, Dec 18 2008