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 A270919 #27 Oct 03 2023 10:36:28
%S A270919 1,2,12,80,552,3912,28224,206208,1520784,11297546,84413912,633713808,
%T A270919 4776117216,36115518376,273868321536,2081866609920,15859616674336,
%U A270919 121046064563376,925411686479820,7085465166635440,54323193841192752,416993869451825424,3204447137019290944
%N A270919 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^n.
%C A270919 From _Peter Bala_, Apr 18 2023: (Start)
%C A270919 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
%C A270919 Conjecture: the supercongruence a(p) == 2*p + 2 (mod p^2) holds for all primes p. Cf. A291697. (End)
%H A270919 Seiichi Manyama, <a href="/A270919/b270919.txt">Table of n, a(n) for n = 0..1118</a> (terms 0..500 from Vaclav Kotesovec)
%F A270919 a(n) ~ c * d^n / sqrt(n), where d = 7.862983395705905261519347909953827161057584... and c = 0.299856802806668079413694689903953367699319...
%F A270919 a(n) = [x^n] 1/theta_4(x)^n, where theta_4() is the Jacobi theta function. - _Ilya Gutkovskiy_, Nov 03 2017
%t A270919 Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
%t A270919 Table[SeriesCoefficient[(QPochhammer[-1, x]/QPochhammer[x, x])^n, {x, 0, n}]/2^n, {n, 0, 25}]
%t A270919 (* Calculation of constants {d,c}: *) eq = FindRoot[{2*s*QPochhammer[r*s] == QPochhammer[-1, r*s], (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/Log[r*s] + r*((Derivative[0, 1][QPochhammer][-1, r*s] - 2*s*Derivative[0, 1][QPochhammer][r*s, r*s]) / (2*QPochhammer[r*s])) == 1}, {r, 1/8}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[(1 - r*s)*Log[r*s]^2*(QPochhammer[r*s] / (Pi*(-r*s*(-1 + r*s) * Log[r*s]*(4*(2*ArcTanh[1 - 2*r*s] + QPolyGamma[0, 1, r*s])* Derivative[0, 1][QPochhammer][r*s, r*s] + r*Log[r*s]*(Derivative[0, 2][QPochhammer][-1, r*s] - 2*s*Derivative[0, 2][QPochhammer][r*s, r*s])) + 2*QPochhammer[r*s] * (4*r*s*ArcTanh[1 - 2*r*s] + 2*(-1 + (-1 + r*s)*ArcTanh[1 - 2*r*s])*Log[1 - r*s] - (-1 + r*s)*(-2 + Log[r*s] - 2*Log[1 - r*s])*QPolyGamma[0, 1, r*s] + (-1 + r*s) * QPolyGamma[0, 1, r*s]^2 + (-1 + r*s)*(QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s]* Derivative[0, 0, 1][QPolyGamma][0, 1, r*s])))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* _Vaclav Kotesovec_, Oct 03 2023 *)
%Y A270919 Main diagonal of A288515.
%Y A270919 Cf. A008485, A008705, A156616, A270913, A270923, A270924, A291697, A309986, A362408.
%K A270919 nonn
%O A270919 0,2
%A A270919 _Vaclav Kotesovec_, Mar 25 2016