A229326 Total sum of 4th powers of parts in all partitions of n.

0, 1, 18, 101, 392, 1119, 2904, 6407, 13578, 26218, 49218, 86782, 150860, 249723, 408810, 647170, 1013278, 1545029, 2337738, 3460218, 5086658, 7350874, 10549872, 14929931, 21009874, 29205500, 40385036, 55289000, 75309056, 101692923, 136710130, 182377824

Offset: 0

Author

Alois P. Heinz, Sep 20 2013

Keywords

• nonn

Comments

The bivariate g.f. for the partition statistic "sum of 4th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^4}*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

Convolution of A001159 and A000041. - 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=4 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^4])(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^4*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^4, {k, 1, n}]; Array[a, 32, 0] (* Jean-François Alcover, Dec 15 2016 *)
Table[Sum[DivisorSigma[4, k]*PartitionsP[n-k], {k, 1, n}], {n, 0, 40}] (* Vaclav Kotesovec, May 27 2018 *)

Formula

a(n) = Sum_{k=1..n} A066633(n,k) * k^4.

G.f.: g(x) = (Sum_{k>=1} k^4*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ 216*sqrt(2)*Zeta(5)/Pi^5 * exp(Pi*sqrt(2*n/3)) * n^(3/2). - Vaclav Kotesovec, May 28 2018