A105124 Three-dimensional small Schroeder numbers.

1, 1, 11, 197, 4593, 126289, 3888343, 130016393, 4629617873, 173225211953, 6746427428131, 271578345652109, 11240106619304609, 476332107976984545, 20601333127791572143, 906951532759564554769, 40554743852511698293601

Offset: 0

Author

N. J. A. Sloane, Apr 09 2005

Keywords

• nonn
• easy

Comments

a(n) = number of increasing tableaux of shape (n,n,n). An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - Oliver Pechenik, May 03 2014

Links

• R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp. (see page 16).

Crossrefs

Cf. A088594.

Programs

PARI

{alias(C,binomial); a(n)=if(n==0,1,sum(k=0,2*n-2, 2^k*sum(j=0,k, 2*(-1)^(k-j)*C(3*n+1,k-j)*C(n+j,n)*C(n+j+1,n)*C(n+j+2,n)/(n+1)^2/(n+2))))} \\ Hanna

Formula

From Paul D. Hanna, Apr 19 2005: (Start)

a(n) = A088594(n)/4 for n>0.

a(0)=1, a(n) = Sum_{k=0..2*n-2} 2^k*Sum_{j=0..k} 2*(-1)^(k-j)*C(3*n+1, k-j)*C(n+j, n)*C(n+j+1, n)*C(n+j+2, n)/(n+1)^2/(n+2) (Sulanke). (End)

Extensions

More terms from Paul D. Hanna, Apr 19 2005