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Number of partitions of n such that (smallest part) = 4*(number of parts).
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%I #16 Jan 24 2022 04:42:40

%S 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,

%U 15,17,17,19,20,22,23,26,27,30,32,35,37,41,43,47,50,54,57,62,65,70,74,79,83,89,93,99,104

%N Number of partitions of n such that (smallest part) = 4*(number of parts).

%F G.f.: Sum_{k>=1} x^(4*k^2)/Product_{j=1..k-1} (1-x^j).

%F a(n) ~ (1 - alfa) * exp(2*sqrt(n*(4*log(alfa)^2 + polylog(2, 1 - alfa)))) * (4*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(8 - 7*alfa) * n^(3/4)), where alfa = 0.8116523200278026483934188589034567041719182934245... is positive real root of the equation alfa^8 + alfa - 1 = 0. - _Vaclav Kotesovec_, Jan 22 2022

%o (PARI) my(N=99, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, k-1, 1-x^j))))

%Y Column 4 of A350890.

%Y Cf. A168656.

%K nonn

%O 1,36

%A _Seiichi Manyama_, Jan 21 2022