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%I #21 Jul 19 2017 19:44:53
%S 1,12,690,52808,4558410,420421056,40433534960,4002511248720,
%T 404653074076602,41573640435563720,4325688482694408060,
%U 454713687334494619200,48204482093235945250800,5146506898529612988887424,552782991828545241240684480,59682974236253934536767852960
%N Number of solid standard Young tableaux of shape [[3*n,n],[n]].
%H Alois P. Heinz, <a href="/A215686/b215686.txt">Table of n, a(n) for n = 0..400</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: 144*n^2*(2*n-1)^2*(3*n-1)*(3*n+1)*(6*n-1)*(6*n+1)*(1054762104*n^8 - 12255641802*n^7 + 61733195125*n^6 - 176088471031*n^5 + 310993546391*n^4 - 348002018003*n^3 + 240698328276*n^2 - 93950633268*n + 15815567520)*a(n) = 3*(16185096657265536*n^16 - 237611988564522912*n^15 + 1583108012261189688*n^14 - 6336788565970939206*n^13 + 16995962918085979601*n^12 - 32239020575894967680*n^11 + 44456039640264512829*n^10 - 45040072648450035120*n^9 + 33379794951068698383*n^8 - 17660683589724361536*n^7 + 6240753949747677391*n^6 - 1170756653568973234*n^5 - 67328664931986180*n^4 + 105177779152514568*n^3 - 28053879047153568*n^2 + 3460226308012800*n - 166227489792000)*a(n-1) - 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(1065259096459008*n^12 - 11533800266013504*n^11 + 53860225549987304*n^10 - 142107589766231326*n^9 + 232900286313689643*n^8 - 245074012300774359*n^7 + 164357596249809711*n^6 - 65773579693921743*n^5 + 11793083348968270*n^4 + 1470831449279884*n^3 - 1161186153316104*n^2 + 208213965227520*n - 12225693916800)*a(n-2) + 97200*(5*n-14)*(5*n-13)*(5*n-12)*(5*n-11)*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(1054762104*n^8 - 3817544970*n^7 + 5477041423*n^6 - 3991100299*n^5 + 1435002321*n^4 - 129425495*n^3 - 68369752*n^2 + 19574652*n - 1364688)*a(n-3). - _Vaclav Kotesovec_, Aug 31 2014
%F a(n) ~ sqrt(330*sqrt(33)-1770)/36 * 5^(5*n) / (Pi * n * 3^(3*n)). - _Vaclav Kotesovec_, Aug 31 2014
%p b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y), `if`(z>x, 0,
%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(3*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[3n, n, n]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y Column k=3 of A176129.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Aug 20 2012
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