A292194 Sum of n-th powers of products of terms in all partitions of n.

1, 1, 5, 36, 610, 13225, 1173652, 92137513, 27960729094, 14612913824364, 11885159817456154, 23676862215173960082, 144210774157588042096815, 778807208565930895328294712, 15863318347221014170216633451982, 908978343753718115412387406378667615

Offset: 0

Author

Seiichi Manyama, Sep 11 2017

Keywords

• nonn

Examples

			5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1.
So a(5) = 5^5 + (4*1)^5 + (3*2)^5 + (3*1*1)^5 + (2*2*1)^5 + (2*1*1*1)^5 + (1*1*1*1*1)^5 = 13225.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..79

Crossrefs

Main diagonal of A292193.

Cf. A292190.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 11 2017

Mathematica

nmax = 20; Table[SeriesCoefficient[Product[1/(1 - k^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 15 2017 *)

PARI

{a(n) = polcoeff(1/prod(k=1, n, 1-k^n*x^k+x*O(x^n)), n)}

Formula

a(n) = [x^n] Product_{k=1..n} 1/(1 - k^n*x^k).

From Vaclav Kotesovec, Sep 15 2017: (Start)

a(n) ~ 3^(n^2/3) if mod(n,3)=0

a(n) ~ 3^(n*(n-4)/3)*2^(2*n+1) if mod(n,3)=1

a(n) ~ 3^(n*(n-2)/3)*2^n if mod(n,3)=2

(End)