A023880 - OEIS

%I #36 Sep 08 2022 08:44:48

%S 1,1,5,32,298,3531,51609,894834,17980052,410817517,10518031721,

%T 298207687029,9273094072138,313757506696967,11474218056441581,

%U 450961669608632160,18954582520550896213,848384721904740036422,40285256621556957160307,2022695276960566890383148

%N Number of partitions in expanding space.

%C 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

%H Alois P. Heinz, <a href="/A023880/b023880.txt">Table of n, a(n) for n = 0..300</a>

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

%F a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 5*exp(-2))/n^2). - _Vaclav Kotesovec_, Mar 14 2015

%F a(n) = (1/n)*Sum_{k=1..n} A283498(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 11 2017

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       add(d*d^d, d=divisors(j)) *a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Feb 04 2015

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

%o (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

%o (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

%o (SageMath) # uses[EulerTransform from A166861]

%o b = EulerTransform(lambda n: n^n)

%o print([b(n) for n in range(20)]) # _Peter Luschny_, Nov 11 2020

%K nonn

%O 0,3

%A _Olivier Gérard_