A229329 Total sum of 7th powers of parts in all partitions of n.

0, 1, 130, 2319, 18962, 99407, 400620, 1323441, 3835406, 9924912, 23736846, 52729348, 111173790, 222415631, 428578374, 794363760, 1430855958, 2500747293, 4274146464, 7130355736, 11681752260, 18764913468, 29690460150, 46211761397, 71016916110, 107622522692

Offset

0,3

Comments

The bivariate g.f. for the partition statistic "sum of 7th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^7}*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^7.

G.f.: g(x) = (Sum_{k>=1} k^7*x^k/(1-x^k))/Product_{q>=1}(1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ 288*sqrt(3)/5 * exp(Pi*sqrt(2*n/3)) * n^3. - 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^7])(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^7*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^7, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

Crossrefs

Column k=7 of A213191.

Sequence in context: A254924 A185584 A301545 * A262108 A250212 A239094

Adjacent sequences: A229326 A229327 A229328 * A229330 A229331 A229332

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2013

Status

approved