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

0,3

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.

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

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 *)

Crossrefs

Column k=3 of A213191.

Sequence in context: A188480 A059722 A267748 * A074225 A055514 A055233

Adjacent sequences: A229322 A229323 A229324 * A229326 A229327 A229328

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2013

Status

approved