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A215123 Number of solid standard Young tableaux of shape [[n^2,n],[n]].

%I A215123 #26 Jul 19 2017 20:10:14
%S A215123 1,2,174,52808,31497284,31113230148,46190668836656,96484621769643360,
%T A215123 270280816277448460968,979042561410295182717884,
%U A215123 4456728497956906393963534248,24916868994347706845906490576432,167903137478620963997932010166057408
%N A215123 Number of solid standard Young tableaux of shape [[n^2,n],[n]].
%H A215123 Alois P. Heinz, <a href="/A215123/b215123.txt">Table of n, a(n) for n = 0..100</a>
%H A215123 S. B. Ekhad, D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229v1 [math.CO], 2012
%H A215123 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A215123 a(n) ~ exp(2*n+2) * n^(2*n-1) / (2*Pi). - _Vaclav Kotesovec_, Jan 19 2015
%p A215123 b:= proc(x, y, z) option remember; `if`(z<y, b(x, z, y),
%p A215123       `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
%p A215123       `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
%p A215123     end:
%p A215123 a:= n-> b(n^2, n, n):
%p A215123 seq(a(n), n=0..15);
%t A215123 $RecursionLimit = 1000; b[x_, y_, z_] :=  b[x, y, z] = If[z<y, b[x, z, y],  If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; a[n_] := b[n^2, n, n]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y A215123 Central row elements of A215122.
%Y A215123 Main diagonal of A176129.
%K A215123 nonn
%O A215123 0,2
%A A215123 _Alois P. Heinz_, Aug 03 2012