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A363096 Number of partitions of n whose least part is a multiple of 5.

%I A363096 #21 May 21 2023 10:20:35
%S A363096 0,0,0,0,1,0,0,0,0,2,1,1,1,1,3,2,3,3,4,7,7,8,10,11,15,16,19,22,27,34,
%T A363096 39,46,54,63,76,86,101,117,136,161,186,214,249,287,335,384,445,509,
%U A363096 588,677,776,888,1020,1163,1334,1519,1735,1975,2253,2564,2917,3312,3762,4265,4842,5477,6203,7012,7928
%N A363096 Number of partitions of n whose least part is a multiple of 5.
%C A363096 In general, for m > 0, if g.f. = Sum_{k>=1} x^(m*k)/Product_{j>=m*k} (1-x^j), then a(n) ~ Pi^(m-1) * (m-1)! * exp(Pi*sqrt(2*n/3)) / (2^(3/2) * 6^(m/2) * n^((m+1)/2)) * (1 - (m*(m+1)/(4*Pi) + (6*m^2 + 18*m + 1 + c)*Pi/144)/sqrt(n/6)), where c = 0 for m > 1 and c = -24 for m = 1. - _Vaclav Kotesovec_, May 21 2023
%H A363096 Vaclav Kotesovec, <a href="/A363096/b363096.txt">Table of n, a(n) for n = 1..10000</a>
%F A363096 G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=5*k} (1-x^j).
%F A363096 a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (2*3^(3/2)*n^3) * (1 - (15*sqrt(6)/(2*Pi) + 241*Pi*sqrt(6)/144) / sqrt(n)). - _Vaclav Kotesovec_, May 21 2023
%t A363096 nmax = 60; Rest[CoefficientList[Series[Sum[x^(5*k)/QPochhammer[x^(5*k), x], {k, 1, nmax/5}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, May 20 2023 *)
%o A363096 (PARI) my(N=70, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=5*k, N, 1-x^j))))
%Y A363096 Cf. A026805, A363094, A363095.
%Y A363096 Cf. A363047.
%K A363096 nonn
%O A363096 1,10
%A A363096 _Seiichi Manyama_, May 19 2023