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A023880
Number of partitions in expanding space.
10
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
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
KEYWORD
nonn
STATUS
approved