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 A376943 #14 Oct 10 2024 05:34:34
%S A376943 1,0,2,2,0,0,4,4,4,4,0,0,8,8,8,16,8,8,8,0,16,16,16,32,32,32,32,32,16,
%T A376943 16,48,32,32,64,64,96,96,96,96,96,96,64,128,96,96,160,128,192,256,256,
%U A376943 256,320,320,320,320,256,384,384,320,384,384,448,576,704,640,768,896
%N A376943 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) * Product_{j=1..k} (1 + x^j).
%C A376943 In general, if d >= 1, b > 0 and g.f. = Sum_{k>=0} d^k * x^(b*k^2 + c*k) * Product_{j=1..k} (1 + x^j), then a(n) ~ r^c * (1+r) * exp(sqrt((2*log(d)^2 + 8*b*log(d)*log(r) + 4*b*(2*b+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*b*(1+r))*n)), where r is the smallest positive real root of the equation d*r^(2*b)*(1+r) = 1.
%H A376943 Vaclav Kotesovec, <a href="/A376943/b376943.txt">Table of n, a(n) for n = 0..10000</a>
%F A376943 a(n) ~ r * (1+r) * exp(sqrt((2*log(2)^2 + 8*log(2)*log(r) + 12*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((3*r + 2)*n)), where r = ((46 - 6*sqrt(57))^(1/3) + (46 + 6*sqrt(57))^(1/3) - 2)/6 is the real root of the equation 2*r^2*(1+r) = 1 (A273065).
%t A376943 nmax = 80; CoefficientList[Series[Sum[2^k * x^(k*(k+1)) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%t A376943 nmax = 80; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^(2*k), {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]
%Y A376943 Cf. A273065, A333179, A376944, A376945, A376947.
%K A376943 nonn
%O A376943 0,3
%A A376943 _Vaclav Kotesovec_, Oct 10 2024