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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2013
STATUS
approved