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 A377081 #13 Oct 16 2024 08:10:57
%S A377081 0,0,0,1,1,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,1,1,1,
%T A377081 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,1,1,1,0,0,0,0,0,0,0,0,0,
%U A377081 0,0,0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,3,2
%N A377081 G.f.: Sum_{k>=1} x^(3*k^2) * Product_{j=1..k} (1 + x^j).
%C A377081 In general, if m > 0 and g.f. = Sum_{k>=1} x^(m*k^2) * Product_{j=1..k} (1 + x^j), then a(n) ~ (1+r) * exp(sqrt((4*m*(2*m+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*m*(1+r))*n)), where r is the smallest positive real root of the equation r^(2*m)*(1+r) = 1.
%H A377081 Vaclav Kotesovec, <a href="/A377081/b377081.txt">Table of n, a(n) for n = 0..10000</a>
%H A377081 Vaclav Kotesovec, <a href="/A377081/a377081.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A377081 a(n) ~ (1+r) * exp(sqrt((84*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((6 + 7*r)*n)), where r = A230154 = 0.898653712628699293260875722... is the real root of the equation r^6*(1+r) = 1.
%t A377081 nmax = 200; CoefficientList[Series[Sum[x^(3*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/3]}], {x, 0, nmax}], x]
%Y A377081 Cf. A306734 (m=1), A377080 (m=2).
%Y A377081 Cf. A230154, A350890, A350894.
%K A377081 nonn,look
%O A377081 0,31
%A A377081 _Vaclav Kotesovec_, Oct 15 2024