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A218433 Sum of the 6th powers of the numbers of standard Young tableaux over all partitions of n.

%I A218433 #13 May 18 2017 03:48:16
%S A218433 1,1,2,66,1524,86100,19902600,3965056200,976304082600,384973061999400,
%T A218433 347437227718904400,365434181398477976400,390696545168036224840800,
%U A218433 475968229571639505471170400,784642922815221782474131569600,2070759893211522247088843511422400
%N A218433 Sum of the 6th powers of the numbers of standard Young tableaux over all partitions of n.
%H A218433 Alois P. Heinz, <a href="/A218433/b218433.txt">Table of n, a(n) for n = 0..60</a>
%H A218433 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p A218433 h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
%p A218433        +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A218433     end:
%p A218433 g:= proc(n, i, l) `if`(n=0, h(l)^6, `if`(i<1, 0, g(n, i-1, l)+
%p A218433       `if`(i>n, 0, g(n-i, i, [l[], i]))))
%p A218433     end:
%p A218433 a:= n-> `if`(n=0, 1, g(n, n, [])):
%p A218433 seq(a(n), n=0..20);
%t A218433 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 A218433 g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l]^6, If[i < 1, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]];
%t A218433 a[n_] := If[n == 0, 1, g[n, n, {}]];
%t A218433 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 18 2017, translated from Maple *)
%Y A218433 Column k=6 of A208447.
%K A218433 nonn
%O A218433 0,3
%A A218433 _Alois P. Heinz_, Oct 28 2012