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

0,5

Comments

A Galton triangle. Essentially the same as A187075.

Links

Table of n, a(n) for n=0..44.

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.

Example

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.

Sequence in context: A261275 A140326 A261781 * A256064 A126436 A102394

Adjacent sequences: A211399 A211400 A211401 * A211403 A211404 A211405

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Feb 09 2013

Status

approved