OFFSET
0,3
COMMENTS
The bivariate g.f. for the partition statistic "sum of 6th powers the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^6}*x^k). The g.f. g at the Formula section has been 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..1000
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^6.
G.f.: g(x) = (Sum_{k>=1} k^6*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 38880*sqrt(2)*Zeta(7)/Pi^7 * exp(Pi*sqrt(2*n/3)) * n^(5/2). - 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^6])(b(n, i-1), b(n-i, i)))))
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^6*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015
MATHEMATICA
(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n - k, k] + PartitionsP[n - k]; T[_, _] = 0; a[n_] := Sum[T[n, k]*k^6, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2013
STATUS
approved