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

1, 1, 2, 10, 64, 596, 8056, 130432, 2534960, 59822884, 1718480368, 56754444440, 2110577206816, 87981286785328, 4129351961475872, 218382856010529472, 12813477368159567200, 822337333595479929044, 57213666993723455063392, 4305630141314873304140008

Offset: 0

Author

David A. Madore, Jul 03 2007

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000041, A000085, A000142.

Column k=3 of A208447.

Programs

Mathematica

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}]];
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]]]]];
a[n_] := If[n == 0, 1, g[n, n, 3, {}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 18 2017, after Alois P. Heinz *)

Formula

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.

Extensions

More terms from Alois P. Heinz, Feb 26 2012