A252761 Total sum of n-th powers of parts in all partitions of n.

0, 1, 6, 39, 392, 4775, 73920, 1323441, 27530298, 644749920, 16877063274, 486936848068, 15373069624220, 526779275391863, 19477946814752586, 772859962684631760, 32758854443379036238, 1477205045259973740909, 70613293111837146235770, 3566735926987461858837256

Offset: 0

Author

Alois P. Heinz, Dec 21 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Main diagonal of A213191.

Cf. A066633.

Programs

Maple

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^k*m]))
          (b(n-p*m, p-1, k)), m=0..n/p)))
    end:
a:= n-> b(n$3)[2]:
seq(a(n), n=0..20);

Mathematica

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m == 0, l, l + {0, l[[1]]*p^k*m}]][b[n - p*m, p - 1, k]], {m, 0, n/p}]]]; a[n_] := b[n, n, n][[2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 07 2017, translated from Maple *)

Formula

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

a(n) ~ c * n^n, where c = 1/QPochhammer(exp(-1)) = 1.98244090741287370368568246556131201568288277843252568635840026086375046496... - Vaclav Kotesovec, May 28 2018, updated Jul 21 2018