A229325 Total sum of cubes of parts in all partitions of n.

0, 1, 10, 39, 122, 287, 660, 1281, 2486, 4392, 7686, 12628, 20790, 32471, 50694, 76560, 115038, 168333, 245784, 350896, 499620, 699468, 975150, 1341077, 1838550, 2490092, 3361260, 4494084, 5986750, 7909231, 10416300, 13616768, 17745948, 22983345, 29672974

Offset: 0

Author

Alois P. Heinz, Sep 20 2013

Keywords

• nonn

Comments

The bivariate g.f. for the partition statistic "sum of cubes of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^3}*x^k). The g.f. g given in the Formula section was obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..8500
• Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Crossrefs

Column k=3 of A213191.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^3])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);

Mathematica

Table[Total[Flatten[IntegerPartitions[n]^3]],{n,0,40}] (* Harvey P. Dale, May 01 2016 *)
b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, If[i>n, b[n, i-1], Function[{g, h}, g + h + {0, h[[1]]*i^3}][b[n, i-1], b[n-i, i]]]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} A066633(n,k) * k^3.

G.f.: g(x) = (Sum_{k>=1} k^3*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ sqrt(3)/5 * exp(Pi*sqrt(2*n/3)) * n. - Vaclav Kotesovec, May 28 2018