A376812 - OEIS

%I #11 Oct 09 2024 16:35:22

%S 1,1,2,2,2,3,5,5,5,7,8,10,13,14,16,19,21,25,29,33,40,45,50,57,64,72,

%T 81,93,104,117,134,148,165,185,204,227,253,280,310,345,381,422,469,

%U 514,567,625,685,753,825,903,990,1086,1186,1297,1419,1548,1692,1845,2007

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

%H Vaclav Kotesovec, <a href="/A376812/b376812.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 1/(4*sqrt(3/2 - 2*sinh(arcsinh(3^(3/2)/2)/3)/sqrt(3))) = 0.27647151570071656262813536...

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

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

%Y Cf. A053261, A376626, A376627, A376813, A216222.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Oct 05 2024