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%I #16 Jan 24 2022 04:43:07
%S 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,
%U 12,13,13,14,14,15,15,16,16,17,17,18,18,19,20,21,22,24,25,27,29,31,33,36,38,41,44,47
%N Number of partitions of n such that (smallest part) = 5*(number of parts).
%F G.f.: Sum_{k>=1} x^(5*k^2)/Product_{j=1..k-1} (1-x^j).
%F a(n) ~ (1 - alfa) * exp(2*sqrt(n*(5*log(alfa)^2 + polylog(2, 1 - alfa)))) * (5*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(10 - 9*alfa) * n^(3/4)), where alfa = 0.8350790427235590476091499923248865165628469558282... is positive real root of the equation alfa^10 + alfa - 1 = 0. - _Vaclav Kotesovec_, Jan 22 2022
%o (PARI) my(N=99, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, k-1, 1-x^j))))
%Y Column 5 of A350890.
%Y Cf. A168656.
%K nonn
%O 1,45
%A _Seiichi Manyama_, Jan 21 2022
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