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

%I A218432 #16 May 18 2017 02:50:23
%S A218432 1,1,2,34,520,16076,1379176,120097552,12801080384,2000907273220,
%T A218432 548936782370416,179067386842497176,62826358527724433632,
%U A218432 25400850150874996376944,12937006577192667715178720,9081992531456407951744097536,7967213735571969862638061300096
%N A218432 Sum of the 5th powers of the numbers of standard Young tableaux over all partitions of n.
%H A218432 Alois P. Heinz, <a href="/A218432/b218432.txt">Table of n, a(n) for n = 0..60</a>
%H A218432 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p A218432 h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
%p A218432        +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A218432     end:
%p A218432 g:= proc(n, i, l) `if`(n=0, h(l)^5, `if`(i<1, 0, g(n, i-1, l)+
%p A218432       `if`(i>n, 0, g(n-i, i, [l[], i]))))
%p A218432     end:
%p A218432 a:= n-> `if`(n=0, 1, g(n, n, [])):
%p A218432 seq(a(n), n=0..20);
%t A218432 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}]];
%t A218432 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]]]]];
%t A218432 a[n_] := If[n == 0, 1, g[n, n, {}]];
%t A218432 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 18 2017, translated from Maple *)
%Y A218432 Column k=5 of A208447.
%K A218432 nonn
%O A218432 0,3
%A A218432 _Alois P. Heinz_, Oct 28 2012