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A320754 Number of partitions of n with eight kinds of 1.

%I A320754 #12 Sep 08 2022 08:46:23
%S A320754 1,8,37,129,376,966,2258,4902,10026,19520,36459,65721,114877,195454,
%T A320754 324706,528069,842531,1321214,2039553,3103562,4660814,6914927,
%U A320754 10144558,14728160,21176077,30171935,42625765,59741868,83105140,114790422,157500479,214739450
%N A320754 Number of partitions of n with eight kinds of 1.
%H A320754 Alois P. Heinz, <a href="/A320754/b320754.txt">Table of n, a(n) for n = 0..10000</a>
%F A320754 G.f.: 1/(1-x)^8 * 1/Product_{j>1} (1-x^j).
%F A320754 Euler transform of 8,1,1,1,... .
%F A320754 a(n) ~ 2^(3/2) * 3^3 * n^(5/2) * exp(Pi*sqrt(2*n/3)) / Pi^7. - _Vaclav Kotesovec_, Oct 24 2018
%p A320754 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A320754       (numtheory[sigma](j)+7)*a(n-j), j=1..n)/n)
%p A320754     end:
%p A320754 seq(a(n), n=0..40);
%t A320754 nmax = 50; CoefficientList[Series[1/((1-x)^7 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 24 2018 *)
%o A320754 (PARI) x='x+O('x^40); Vec(1/((1-x)^8*prod(j=2, 40, 1-x^j))) \\ _G. C. Greubel_, Oct 27 2018
%o A320754 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^8*(&*[1-x^j: j in [2..30]])))); // _G. C. Greubel_, Oct 27 2018
%Y A320754 Column k=8 of A292508.
%K A320754 nonn
%O A320754 0,2
%A A320754 _Alois P. Heinz_, Oct 20 2018