A229331 Total sum of 9th powers of parts in all partitions of n.

0, 1, 514, 20199, 283370, 2256695, 12637956, 55247745, 202345886, 644749920, 1846772550, 4836548836, 11795957334, 27022021703, 58819382790, 122237638440, 244429962966, 471615005229, 882955864560, 1606698758560, 2853601781340, 4952029001892, 8423307325854

Offset

0,3

Comments

The bivariate g.f. for the partition statistic "sum of 9th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^9}*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^9.

G.f.: g(x) = (Sum_{k>=1} k^9*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ 27648*sqrt(3)/11 * exp(Pi*sqrt(2*n/3)) * n^4. - 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^9])(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^9*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^9, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

Crossrefs

Column k=9 of A213191.

Sequence in context: A213065 A045054 A301547 * A253637 A231415 A271760

Adjacent sequences: A229328 A229329 A229330 * A229332 A229333 A229334

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2013

Status

approved