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 A294591 #41 Feb 16 2025 08:33:51
%S A294591 1,1,3,8,18,40,88,184,384,783,1573,3110,6087,11745,22450,42466,79597,
%T A294591 147890,272632,498696,905846,1634270,2929804,5220581,9249440,16297659,
%U A294591 28567571,49825296,86487331,149438681,257077485,440378787,751313413,1276765557,2161511352
%N A294591 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(3*k-1)/2)*(1 - x^(2*k))^(k*(3*k+1)/2)).
%C A294591 Euler transform of the generalized pentagonal numbers (A001318).
%H A294591 Vaclav Kotesovec, <a href="/A294591/b294591.txt">Table of n, a(n) for n = 0..5000</a>
%H A294591 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A294591 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>
%F A294591 G.f.: Product_{k>=1} 1/(1 - x^k)^A001318(k).
%F A294591 a(n) ~ exp(Pi * 2^(5/4) / (3*5^(1/4)) * n^(3/4) + 3*Zeta(3) * sqrt(5*n) / (2^(3/2) * Pi^2) + (Pi/48 - 45*Zeta(3)^2 / (8*Pi^5)) * (5*n/2)^(1/4) + 225*Zeta(3)^3 / (8*Pi^8) - 11*Zeta(3) / (64*Pi^2))/ (2^(95/48) * 5^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Nov 07 2017
%t A294591 nmax = 34; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (3 k - 1)/2) (1 - x^(2 k))^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A294591 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]
%Y A294591 Cf. A000294, A001318, A095699, A278768, A294654, A294655, A294667, A294669.
%K A294591 nonn
%O A294591 0,3
%A A294591 _Ilya Gutkovskiy_, Nov 05 2017