A350892 - OEIS

%I #25 Oct 14 2024 13:07:37

%S 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 115,135,158,184,214,248,286,330,379,435,497,569,648,739,840,955,1082,

%U 1228,1388,1572,1775,2005,2259,2549,2867,3228,3626,4076,4571,5131,5745,6438,7199,8053,8992,10045,11199

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

%H Vaclav Kotesovec, <a href="/A350892/b350892.txt">Table of n, a(n) for n = 1..20000</a>

%F G.f.: Sum_{k>=1} x^(3*k^2)/Product_{j=1..3*k-1} (1-x^j).

%F 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 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 Table[Count[IntegerPartitions[n],_?(3#[[-1]]==Length[#]&)],{n,70}] (* _Harvey P. Dale_, Jul 13 2023 *)

%o (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 Column 3 of A350889.

%Y Cf. A168657, A237756, A350894.

%K nonn

%O 1,5

%A _Seiichi Manyama_, Jan 21 2022