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A130721 Sum of the cubes of the number of standard Young tableaux over all partitions of n.

%I A130721 #18 May 18 2017 03:23:14
%S A130721 1,1,2,10,64,596,8056,130432,2534960,59822884,1718480368,56754444440,
%T A130721 2110577206816,87981286785328,4129351961475872,218382856010529472,
%U A130721 12813477368159567200,822337333595479929044,57213666993723455063392,4305630141314873304140008
%N A130721 Sum of the cubes of the number of standard Young tableaux over all partitions of n.
%C A130721 The sum of the zeroth power of the number f(p) of standard Young tableaux gives the partition function (A000041), the sum of the first power of f(p) gives the involution function (A000085), the sum of the squares of f(p) gives the factorial function (A000142), so this sequence is the natural one after them.
%H A130721 Alois P. Heinz, <a href="/A130721/b130721.txt">Table of n, a(n) for n = 0..60</a>
%F A130721 For p a partition of n, let f(p) be the number of standard Young tableaux with shape p. Then a(n) = sum(f(p)^3) where the sum ranges over all partitions p of n.
%e A130721 a(4) = 1^3 + 3^3 + 2^3 + 3^3 + 1^3 because the five partitions of 4 (namely 4, 3+1, 2+2, 2+1+1, 1+1+1+1) have respectively 1, 3, 2, 3, 1 standard Young tableaux.
%t A130721 h[l_] := With[{n=Length[l]}, Sum[i, {i, 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 A130721 g[n_, i_, k_, l_] := g[n, i, l, k] = If[n == 0, h[l]^k, If[i < 1, 0, g[n, i - 1, k, l] + If[i > n, 0, g[n - i, i, k, Append[l, i]]]]];
%t A130721 a[n_] := If[n == 0, 1, g[n, n, 3, {}]];
%t A130721 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 18 2017, after _Alois P. Heinz_ *)
%Y A130721 Cf. A000041, A000085, A000142.
%Y A130721 Column k=3 of A208447.
%K A130721 nonn
%O A130721 0,3
%A A130721 _David A. Madore_, Jul 03 2007
%E A130721 More terms from _Alois P. Heinz_, Feb 26 2012