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%I #17 Feb 07 2024 03:38:02
%S 1,1,11,197,4593,126289,3888343,130016393,4629617873,173225211953,
%T 6746427428131,271578345652109,11240106619304609,476332107976984545,
%U 20601333127791572143,906951532759564554769,40554743852511698293601
%N Three-dimensional small Schroeder numbers.
%C 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
%H R. A. Sulanke, <a href="https://doi.org/10.37236/1807">Generalizing Narayana and Schroeder Numbers to Higher Dimensions</a>, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp. (see page 16).
%F From _Paul D. Hanna_, Apr 19 2005: (Start)
%F a(n) = A088594(n)/4 for n>0.
%F 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)
%o (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
%Y Cf. A088594.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Apr 09 2005
%E More terms from _Paul D. Hanna_, Apr 19 2005
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