A229332 Total sum of 10th powers of parts in all partitions of n.

0, 1, 1026, 60077, 1110704, 10936407, 72573360, 365983991, 1513288698, 5365004410, 16877063274, 48105808222, 126584890148, 310963328163, 721354362186, 1590587613754, 3359058693214, 6822189191429, 13396265918970, 25501949210562, 47248199227946, 85355336473378

Offset: 0

Author

Alois P. Heinz, Sep 20 2013

Keywords

• nonn

Comments

The bivariate g.f. for the partition statistic "sum of 10th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^10}*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

In general, column k>0 of A213191 is asymptotic to 2^((k-3)/2) * 3^(k/2) * k! * Zeta(k+1) / Pi^(k+1) * exp(Pi*sqrt(2*n/3)) * n^((k-1)/2). - Vaclav Kotesovec, May 28 2018

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.

Crossrefs

Column k=10 of A213191.

Programs

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^10])(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^10*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^10, {k, 1, n}]; Array[a, 40, 0] (* Jean-François Alcover, Dec 15 2016 *)

Formula

a(n) = Sum_{j=k..n} A066633(n,k) * k^10.

G.f.: g(x) = (Sum_{k>=1} k^10*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ 7054387200*sqrt(2)*Zeta(11)/Pi^11 * exp(Pi*sqrt(2*n/3)) * n^(9/2). - Vaclav Kotesovec, May 28 2018