A394019 - OEIS

%I #13 Mar 13 2026 17:22:52

%S 1,0,0,0,0,1,0,0,1,0,1,0,0,2,0,1,1,0,2,0,2,2,0,2,1,2,3,0,3,3,2,3,1,4,

%T 4,2,4,3,4,5,5,5,5,4,6,7,6,6,7,8,9,6,8,12,9,11,9,10,15,10,15,14,12,17,

%U 13,18,20,13,23,20,20,23,17,29,26,22,29,24,33,32

%N Expansion of g.f.: Product_{k>=1} 1 / (1 - x^(k^2 + 4)).

%C a(n) is the number of partitions of n into parts of the form k^2+4 (for k>=1).

%H Vaclav Kotesovec, <a href="/A394019/b394019.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ zeta(3/2)^(2/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2^(4/3) * sqrt(3) * Pi^(1/6) * sinh(2*Pi) * n^(7/6)) * (1 - (34 + 36*Pi * zeta(1/2) * zeta(3/2)) / (9*2^(5/3) * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3))).

%t nmax = 150; CoefficientList[Series[1/Product[1 - x^(k^2 + 4), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

%Y Cf. A001156, A393990, A394017, A394018, A394020.

%K nonn

%O 0,14

%A _Vaclav Kotesovec_, Mar 06 2026