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A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

%I A350892 #25 Oct 14 2024 13:07:37
%S A350892 0,0,1,1,2,2,3,3,4,4,5,6,7,8,10,12,15,18,22,27,33,40,48,58,69,82,98,
%T A350892 115,135,158,184,214,248,286,330,379,435,497,569,648,739,840,955,1082,
%U A350892 1228,1388,1572,1775,2005,2259,2549,2867,3228,3626,4076,4571,5131,5745,6438,7199,8053,8992,10045,11199
%N A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).
%H A350892 Vaclav Kotesovec, <a href="/A350892/b350892.txt">Table of n, a(n) for n = 1..20000</a>
%F A350892 G.f.: Sum_{k>=1} x^(3*k^2)/Product_{j=1..3*k-1} (1-x^j).
%F A350892 a(n) ~ c * exp(2*sqrt((5*log(A075778)^2 + 2*polylog(2, 1 - A075778))*n)) / n^(3/4), where c = (3*log(A075778)^2 + polylog(2, A075778^2))^(1/4) / (2*sqrt(3*Pi*(1 + A075778)*(2 + 3*A075778))) = 0.0582980106266835787... - _Vaclav Kotesovec_, Jan 24 2022, updated Oct 14 2024
%t A350892 CoefficientList[Series[Sum[x^(3k^2)/Product[1-x^j,{j,3k-1}],{k,64}],{x,0,64}],x] (* _Stefano Spezia_, Jan 22 2022 *)
%t A350892 Table[Count[IntegerPartitions[n],_?(3#[[-1]]==Length[#]&)],{n,70}] (* _Harvey P. Dale_, Jul 13 2023 *)
%o A350892 (PARI) my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, 3*k-1, 1-x^j))))
%Y A350892 Column 3 of A350889.
%Y A350892 Cf. A168657, A237756, A350894.
%K A350892 nonn
%O A350892 1,5
%A A350892 _Seiichi Manyama_, Jan 21 2022