A229327 Total sum of 5th powers of parts in all partitions of n.

0, 1, 34, 279, 1370, 4775, 14196, 35745, 83486, 177120, 358710, 681316, 1257414, 2212343, 3811590, 6344760, 10381686, 16534989, 25994160, 39973360, 60802860, 90875412, 134507694, 196208405, 283895550, 405646460, 575437476, 807778980, 1126478494, 1556675935

Offset

0,3

Comments

The bivariate g.f. for the partition statistic "sum of 5th powers the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^5}*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^5.

G.f.: g(x) = (Sum_{k>=1} k^5*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015

a(n) ~ 16*sqrt(3)/7 * exp(Pi*sqrt(2*n/3)) * n^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^5])(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^5*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

Table[Total[Flatten[IntegerPartitions[n]]^5], {n, 0, 30}] (* Harvey P. Dale, Jun 24 2014 *)
(* 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^5, {k, 1, n}]; Array[a, 45, 0] (* Jean-François Alcover, Dec 15 2016 *)

Crossrefs

Column k=5 of A213191.

Sequence in context: A248076 A301543 A252999 * A209891 A027006 A156531

Adjacent sequences: A229324 A229325 A229326 * A229328 A229329 A229330

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2013

Status

approved