%I #10 Oct 01 2024 05:13:43

%S 1,0,1,2,1,0,1,2,1,2,4,2,2,4,2,2,4,4,5,4,6,10,6,6,9,8,9,8,10,12,11,14,

%T 14,16,17,18,23,20,22,26,26,26,27,34,31,32,39,40,43,42,48,54,55,56,63,

%U 72,68,74,80,84,88,90,101,104,109,112,121,130,132,144,152,160

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

%H Vaclav Kotesovec, <a href="/A376623/b376623.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j-1))^2.

%F a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = sqrt(sinh(arcsinh(3*sqrt(93)/2)/3)) / (sqrt(2)*93^(1/4)) = 0.26678318911398751342...

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

%t nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*(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. A376542, A376580, A376621, A376626, A376629.

%K nonn

%O 0,4

%A _Vaclav Kotesovec_, Sep 30 2024