This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384426 #14 Jun 16 2025 06:11:04
%S A384426 0,1,2,2,3,2,3,3,4,4,4,5,6,5,6,7,8,8,9,9,10,12,12,13,14,14,16,18,19,
%T A384426 20,21,22,24,26,28,30,32,33,34,38,40,43,46,48,51,54,56,60,64,67,72,77,
%U A384426 80,84,88,92,98,105,110,116,122,128,134,142,148,155,164,172
%N A384426 G.f.: Sum_{k>=1} x^k * Product_{j=k..2*k} (1 + x^j).
%H A384426 Vaclav Kotesovec, <a href="/A384426/b384426.txt">Table of n, a(n) for n = 0..10000</a>
%F A384426 a(n) ~ c * exp(r*sqrt(n)) / sqrt(n), where r = 0.926140105877... = 2*sqrt((3/2)*log(z)^2 - polylog(2, 1-z) + polylog(2, 1-z^2)), where z = (-1 + (44 - 3*sqrt(177))^(1/3) + (44 + 3*sqrt(177))^(1/3))/6 = 0.82948354095849703967... is the real root of the equation z^3*(1 - z)/(1 - z^2)^2 = 1 and c = 0.6975701...
%t A384426 nmax = 100; CoefficientList[Series[Sum[x^k * Product[1 + x^j, {j,k,2*k}], {k, 1, nmax}], {x, 0, nmax}], x]
%t A384426 nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 + x^(2*k - 1))*(1 + x^(2*k))/(1 + x^k)]; p = Normal[p + O[x]^nmax]; s += p*(1 + x^k)*x^k;, {k, 1, nmax}]; Take[CoefficientList[s, x], nmax + 1]
%Y A384426 Cf. A237824, A207642.
%K A384426 nonn
%O A384426 0,3
%A A384426 _Vaclav Kotesovec_, Jun 14 2025