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 A377075 #10 Oct 15 2024 05:11:27
%S A377075 0,0,0,0,0,0,0,0,1,1,2,3,5,7,11,15,21,28,38,49,65,82,105,131,164,201,
%T A377075 248,300,364,436,522,618,734,861,1011,1178,1372,1586,1835,2108,2422,
%U A377075 2768,3162,3595,4088,4627,5237,5907,6660,7485,8414,9429,10568,11817,13213
%N A377075 G.f.: Sum_{k>=0} x^(8*k^2) / Product_{j=1..8*k-1} (1 - x^j).
%C A377075 In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m.
%H A377075 Vaclav Kotesovec, <a href="/A377075/b377075.txt">Table of n, a(n) for n = 0..10000</a>
%F A377075 Limit_{n->oo} a(n)^(1/sqrt(n)) = A376658.
%F A377075 a(n) ~ r^2 * (8*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((8*log(r)^2 + polylog(2, r^2))*n)) / (8*sqrt(Pi*(4 - 3*r^2)) * n^(3/4)), where r = 0.8511709340670154789... is the positive real root of the equation r^2 = 1 - r^8.
%t A377075 nmax = 100; CoefficientList[Series[Sum[x^(8*k^2)/Product[1-x^j, {j, 1, 8*k-1}], {k, 1, Sqrt[nmax/8]}], {x, 0, nmax}], x]
%Y A377075 Column 8 of A350889.
%Y A377075 Cf. A376658.
%K A377075 nonn
%O A377075 0,11
%A A377075 _Vaclav Kotesovec_, Oct 15 2024