%I #9 Oct 01 2024 05:22:24

%S 1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,1,1,0,2,2,0,1,1,1,1,1,2,1,2,2,1,2,

%T 1,2,2,1,2,2,2,1,3,3,1,3,3,3,2,2,3,3,3,2,4,3,3,4,3,4,3,4,4,4,4,4,5,4,

%U 5,5,5,4,6,6,4,6,5,7,6,5,7,7,6,6,8,7,7,7,9,8

%N G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^(2*j-1)).

%H Vaclav Kotesovec, <a href="/A376629/b376629.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Sum_{k>=0} Product_{j=1..k} (x^(2*j) + x^(4*j-1)).

%F a(n) ~ exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)).

%t nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1))*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

%t nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

%Y Cf. A003106, A053251, A053258, A333179, A376630.

%K nonn

%O 0,21

%A _Vaclav Kotesovec_, Sep 30 2024