|
%I #18 May 18 2017 03:23:14
%S 1,1,2,10,64,596,8056,130432,2534960,59822884,1718480368,56754444440,
%T 2110577206816,87981286785328,4129351961475872,218382856010529472,
%U 12813477368159567200,822337333595479929044,57213666993723455063392,4305630141314873304140008
%N Sum of the cubes of the number of standard Young tableaux over all partitions of n.
%C 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 Alois P. Heinz, <a href="/A130721/b130721.txt">Table of n, a(n) for n = 0..60</a>
%F 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 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 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 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 a[n_] := If[n == 0, 1, g[n, n, 3, {}]];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 18 2017, after _Alois P. Heinz_ *)
%Y Cf. A000041, A000085, A000142.
%Y Column k=3 of A208447.
%K nonn
%O 0,3
%A _David A. Madore_, Jul 03 2007
%E More terms from _Alois P. Heinz_, Feb 26 2012
|
