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%I #9 Jan 04 2023 21:08:03
%S 1,1,10,276,14318,1214358,150910592,25454753376,5599142988564,
%T 1557618719594808,532482249378122738,218108013886160729600,
%U 105215894641522373026220,59025152558043462549357094,38095446968224172036448488814,27985301641485576224718954999962
%N Number of solid standard Young tableaux of cylindrical shape lambda X 3, where lambda ranges over all partitions of n.
%H S. B. Ekhad and D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229 [math.CO], 2012
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p b:= proc(l) option remember; local m; m:= nops(l);
%p `if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
%p `if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>
%p `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
%p j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
%p end:
%p g:= proc(n, i, l) `if`(n=0 or i=1, b(map(x->[3$x], [l[], 1$n])),
%p add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))
%p end:
%p a:= n-> g(n$2, []):
%p seq(a(n), n=0..10);
%t b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {0} == {}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i + 1]]] < j, 0, l[[i + 1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j + 1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]];
%t g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}]& /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
%t a[n_] := g[n, n, 3, {}]; Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Dec 28 2022, after _Alois P. Heinz_ in A215204 *)
%Y Column k=3 of A215204.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 23 2017
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