A350897 - OEIS

%I #28 Oct 14 2024 14:30:30

%S 0,0,0,0,1,1,2,3,5,6,9,11,15,18,23,27,34,39,47,55,65,74,87,99,115,131,

%T 151,172,199,226,260,298,343,393,454,522,603,696,804,929,1076,1243,

%U 1438,1664,1924,2222,2567,2961,3413,3931,4520,5193,5959,6827,7811,8928,10186,11607,13208,15008,17028,19297

%N Number of partitions of n such that 5*(smallest part) = (number of parts).

%C In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m. - _Vaclav Kotesovec_, Oct 14 2024

%H Vaclav Kotesovec, <a href="/A350897/b350897.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} x^(5*k^2)/Product_{j=1..5*k-1} (1-x^j).

%F a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 0.42067169741517... and c = 0.04778365700734... - _Vaclav Kotesovec_, Jan 26 2022

%F a(n) ~ r^2 * (5*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((5*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(5*Pi*(5 - 3*r^2)) * n^(3/4)), where r = 0.808730600479392... is the real root of the equation r^2 = 1 - r^5. - _Vaclav Kotesovec_, Oct 14 2024

%t CoefficientList[Series[Sum[x^(5k^2)/Product[1-x^j,{j,5k-1}],{k,62}],{x,0,62}],x] (* _Stefano Spezia_, Jan 22 2022 *)

%o (PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, 5*k-1, 1-x^j))))

%Y Column 5 of A350889.

%Y Cf. A168657.

%K nonn

%O 1,7

%A _Seiichi Manyama_, Jan 21 2022