A211402 Triangle T(n,k), 0<=k<=n, given by (0,2,0,4,0,6,0,8,0,10,0,...) DELTA (1, 2, 3, 4, 5, 6, 7, 8, 9,...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 2, 3, 0, 4, 18, 15, 0, 8, 84, 180, 105, 0, 16, 360, 1500, 2100, 945, 0, 32, 1488, 10800, 27300, 28350, 10395, 0, 64, 6048, 72240, 294000, 529200, 436590, 135135, 0, 128, 24384, 463680, 2857680, 7938000, 11060280, 7567560, 2027025

Offset: 0

Author

Philippe Deléham, Feb 09 2013

Keywords

• easy
• nonn
• tabl

Comments

A Galton triangle. Essentially the same as A187075.

Examples

			Triangle begins :
1
0, 1
0, 2, 3
0, 4, 18, 15
0, 8, 84, 180, 105
0, 16, 360, 1500, 2100, 945
0, 32, 1488, 10800, 27300, 28350, 10395
		

Crossrefs

Cf. A000079, A001147, A048993.

Formula

T(n,k) = 2*k*T(n-1,k) + (2*k-1)*T(n-1,k-1), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.

T(n,k) = 2^(n-k)*A001147(k)*A048993(n,k).

G.f.: F(x,t) = 1 + x*t + (2*x+3*x^2)*t^2/2! + (4*x+18*x^2+15*x^3)*t^3/3!+ ... = Sum_{n = 0..inf}R(n,x)*t^n/n!.

The row polynomials R(n,x) satisfy the recursion R(n+1,x) = 2*(x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x.