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A320756 Number of partitions of n with ten kinds of 1.

%I A320756 #12 Sep 08 2022 08:46:23
%S A320756 1,10,56,231,782,2299,6074,14751,33454,71677,146359,286762,542042,
%T A320756 992776,1768216,3071725,5217765,8685019,14191826,22802195,36073378,
%U A320756 56259488,86590156,131648984,197883889,294290729,433323334,632097807,913977420,1310647455,1864817969
%N A320756 Number of partitions of n with ten kinds of 1.
%H A320756 Alois P. Heinz, <a href="/A320756/b320756.txt">Table of n, a(n) for n = 0..10000</a>
%F A320756 G.f.: 1/(1-x)^10 * 1/Product_{j>1} (1-x^j).
%F A320756 Euler transform of 10,1,1,1,... .
%F A320756 a(n) ~ 2^(5/2) * 3^4 * n^(7/2) * exp(Pi*sqrt(2*n/3)) / Pi^9. - _Vaclav Kotesovec_, Oct 24 2018
%p A320756 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A320756       (numtheory[sigma](j)+9)*a(n-j), j=1..n)/n)
%p A320756     end:
%p A320756 seq(a(n), n=0..40);
%t A320756 nmax = 50; CoefficientList[Series[1/((1-x)^9 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 24 2018 *)
%o A320756 (PARI) x='x+O('x^30); Vec(1/((1-x)^10*prod(j=2, 40, 1-x^j))) \\ _G. C. Greubel_, Oct 27 2018
%o A320756 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^10*(&*[1-x^j: j in [2..30]])))); // _G. C. Greubel_, Oct 27 2018
%Y A320756 Column k=10 of A292508.
%K A320756 nonn
%O A320756 0,2
%A A320756 _Alois P. Heinz_, Oct 20 2018