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%I #23 Jul 19 2017 19:40:24
%S 1,6,174,7020,325590,16290708,854630476,46305862488,2568272967270,
%T 144984584562180,8298621602461476,480298712286979560,
%U 28052543639835133692,1650956086756046986440,97790578929910135588440,5824509559447044190027952,348581174512709008160833158
%N Number of solid standard Young tableaux of shape [[2*n,n],[n]].
%H Alois P. Heinz, <a href="/A214801/b214801.txt">Table of n, a(n) for n = 0..555</a>
%H 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F Recurrence: (n-1)*n^2*(2*n-1)*(2*n+1)*(4*n-1)*(4*n+1)*(392*n^4 - 2044*n^3 + 4216*n^2 - 3944*n + 1377)*a(n) = 2*(n-1)*(1859648*n^10 - 13670048*n^9 + 43255264*n^8 - 75152192*n^7 + 75863336*n^6 - 41825576*n^5 + 7317576*n^4 + 5067372*n^3 - 3441344*n^2 + 785094*n - 59535)*a(n-1) - 4*(2*n-3)*(4*n-7)*(4*n-5)*(1310848*n^8 - 7998592*n^7 + 19695952*n^6 - 24269488*n^5 + 15125236*n^4 - 3514192*n^3 - 1066614*n^2 + 533457*n - 45927)*a(n-2) + 5184*n*(2*n-5)*(2*n-3)*(4*n-11)*(4*n-9)*(4*n-7)*(4*n-5)*(392*n^4 - 476*n^3 + 436*n^2 - 76*n - 3)*a(n-3). - _Vaclav Kotesovec_, Aug 31 2014
%F a(n) ~ sqrt((5*sqrt(5)-11)/4) * 64^n / (Pi*n). - _Vaclav Kotesovec_, Aug 31 2014
%p b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),
%p `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
%p `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
%p end:
%p a:= n-> b(2*n, n, n):
%p seq(a(n), n=0..20);
%t 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[2n, n, n]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y Central row elements of A214775.
%Y Column k=2 of A176129.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Jul 28 2012
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