A218432 Sum of the 5th powers of the numbers of standard Young tableaux over all partitions of n.

1, 1, 2, 34, 520, 16076, 1379176, 120097552, 12801080384, 2000907273220, 548936782370416, 179067386842497176, 62826358527724433632, 25400850150874996376944, 12937006577192667715178720, 9081992531456407951744097536, 7967213735571969862638061300096

Offset: 0

Author

Alois P. Heinz, Oct 28 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60
• Wikipedia, Young tableau

Crossrefs

Column k=5 of A208447.

Programs

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) `if`(n=0, h(l)^5, `if`(i<1, 0, g(n, i-1, l)+
      `if`(i>n, 0, g(n-i, i, [l[], i]))))
    end:
a:= n-> `if`(n=0, 1, g(n, n, [])):
seq(a(n), n=0..20);

Mathematica

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l]^5, If[i < 1, 0, g[n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]];
a[n_] := If[n == 0, 1, g[n, n, {}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 18 2017, translated from Maple *)