A023880 Number of partitions in expanding space.

1, 1, 5, 32, 298, 3531, 51609, 894834, 17980052, 410817517, 10518031721, 298207687029, 9273094072138, 313757506696967, 11474218056441581, 450961669608632160, 18954582520550896213, 848384721904740036422, 40285256621556957160307, 2022695276960566890383148

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Comments

Also partitions of n into 1 sort of 1, 4 sorts of 2, 27 sorts of 3, ..., k^k sorts of k. - Joerg Arndt, Feb 04 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^(k^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(
      add(d*d^d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 04 2015

Mathematica

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)

PARI

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^(k^k))) \\ G. C. Greubel, Oct 31 2018

SageMath

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^n)
print([b(n) for n in range(20)]) # Peter Luschny, Nov 11 2020

Formula

G.f.: 1 / Product_{k>=1} (1 - x^k)^(k^k).

a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 5*exp(-2))/n^2). - Vaclav Kotesovec, Mar 14 2015

a(n) = (1/n)*Sum_{k=1..n} A283498(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 11 2017