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A105124
Three-dimensional small Schroeder numbers.
3
1, 1, 11, 197, 4593, 126289, 3888343, 130016393, 4629617873, 173225211953, 6746427428131, 271578345652109, 11240106619304609, 476332107976984545, 20601333127791572143, 906951532759564554769, 40554743852511698293601
OFFSET
0,3
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).
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)
PROG
(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
CROSSREFS
Cf. A088594.
Sequence in context: A243646 A186251 A234628 * A272500 A034831 A142362
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Apr 09 2005
EXTENSIONS
More terms from Paul D. Hanna, Apr 19 2005
STATUS
approved