A393990 - OEIS

%I #17 Mar 13 2026 18:39:02

%S 1,0,1,0,1,1,1,1,1,1,3,1,3,1,3,3,3,4,3,4,6,4,7,4,7,7,8,9,8,9,12,10,14,

%T 10,15,14,17,18,17,19,22,21,26,22,28,27,31,33,32,35,39,39,46,41,50,48,

%U 55,57,57,61,67,67,77,71,83,82,90,95,95,102,109,111,123

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

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

%C In general, if b > 0 and g.f. is Product_{k>=1} 1/(1 - x^(k^2 + b)), then a(n) ~ sqrt(b) * zeta(3/2)^(2/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2^(7/3) * sqrt(3) * Pi^(1/6) * sinh(Pi*sqrt(b)) * n^(7/6)) * (1 - (34 + 9*b*Pi * zeta(1/2) * zeta(3/2)) / (9*2^(5/3) * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3))).

%H Vaclav Kotesovec, <a href="/A393990/b393990.txt">Table of n, a(n) for n = 0..10000</a>

%H Vaclav Kotesovec, <a href="/A393990/a393990_1.jpg">Graph - the asymptotic ratio (200000 terms)</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^(7/3) * sqrt(3) * Pi^(1/6) * sinh(Pi) * n^(7/6)) * (1 - (34 + 9*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+1), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

%Y Cf. A001156, A393991.

%K nonn

%O 0,11

%A _Vaclav Kotesovec_, Mar 06 2026