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A320753 Number of partitions of n with seven kinds of 1.

%I A320753 #13 Sep 08 2022 08:46:23
%S A320753 1,7,29,92,247,590,1292,2644,5124,9494,16939,29262,49156,80577,129252,
%T A320753 203363,314462,478683,718339,1064009,1557252,2254113,3229631,4583602,
%U A320753 6447917,8995858,12453830,17116103,23363272,31685282,42710057,57238971,76290668,101155025
%N A320753 Number of partitions of n with seven kinds of 1.
%H A320753 Alois P. Heinz, <a href="/A320753/b320753.txt">Table of n, a(n) for n = 0..10000</a>
%F A320753 G.f.: 1/(1-x)^7 * 1/Product_{j>1} (1-x^j).
%F A320753 Euler transform of 7,1,1,1,... .
%F A320753 a(n) ~ 2 * 3^(5/2) * n^2 * exp(Pi*sqrt(2*n/3)) / Pi^6. - _Vaclav Kotesovec_, Oct 24 2018
%p A320753 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A320753       (numtheory[sigma](j)+6)*a(n-j), j=1..n)/n)
%p A320753     end:
%p A320753 seq(a(n), n=0..40);
%t A320753 nmax = 50; CoefficientList[Series[1/((1-x)^6 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 24 2018 *)
%o A320753 (PARI) x='x+O('x^30); Vec(1/((1-x)^7*prod(j=2, 40, 1-x^j))) \\ _G. C. Greubel_, Oct 27 2018
%o A320753 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^7*(&*[1-x^j: j in [2..30]])))); // _G. C. Greubel_, Oct 27 2018
%Y A320753 Column k=7 of A292508.
%K A320753 nonn
%O A320753 0,2
%A A320753 _Alois P. Heinz_, Oct 20 2018