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

0,3

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

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

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 *)

Prog

(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
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^(k^k): k in [1..m]]) )); // 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

Crossrefs

Sequence in context: A305305 A331339 A307497 * A104031 A294957 A363397

Adjacent sequences: A023877 A023878 A023879 * A023881 A023882 A023883

Keyword

nonn

Author

Olivier Gérard

Status

approved