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 A274621 #35 Jan 05 2022 05:20:00
%S A274621 1,-2,3,-6,11,-18,28,-44,69,-104,152,-222,323,-460,645,-902,1254,
%T A274621 -1722,2343,-3174,4278,-5722,7601,-10056,13250,-17358,22623,-29382,
%U A274621 38021,-48984,62857,-80404,102528,-130282,165002,-208398,262495,-329666,412878,-515840,642941,-799362,991478
%N A274621 Coefficients in the expansion Product_{ n>=1 } (1-q^(2n-1))^2/(1-q^(2n))^2.
%C A274621 This is the reciprocal of the g.f. for A008441.
%C A274621 From _Wolfdieter Lang_, Jul 05 2016: (Start)
%C A274621 The g.f. is the square of the one for A106507.
%C A274621 Expansion of 1/(k/(4*q^(1/2)) * (2/Pi)*K(k)) in powers of q^2, where k is the modulus (k^2 is the parameter), K is the real quarter period and q is the Jacobi nome of elliptic functions. See a similar Jul 05 2016 comment on A008441. This appears as a factor in the sn and cn formulas of Abramowitz-Stegun. p. 575, 16.23.1 and 16.23.2. (End)
%D A274621 R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105.
%H A274621 Seiichi Manyama, <a href="/A274621/b274621.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)
%H A274621 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
%H A274621 R. W. Gosper, <a href="/A274621/a274621.pdf">Experiments and discoveries in q-trigonometry</a>, Preprint.
%H A274621 R. W. Gosper, <a href="/A274621/a274621_1.pdf">q-Trigonometry: Some Prefatory Afterthoughts</a>
%F A274621 From _Wolfdieter Lang_, Jul 05 2016: (Start)
%F A274621 G.f.: 1/(theta_2(0, sqrt(q))/(2*q^(1/8)))^2, with the Jacobi theta_2 function.
%F A274621 G.f.: 1/(Sum_{n >= 0} q^(n*(n+1)/2))^2.
%F A274621 G.f.: 1/(Prod_{n >= 1} (1 - q^n) * (1 + q^n)^2)^2 = 1/(Prod_{n >= 1} (1 - q^(2*n)) * (1 + q^n ))^2 = Prod_{n >= 1} (1 - q^(2n-1))^2 / (1 - q^(2n))^2. For the last equality, giving the g.f. of the name, see the Euler identity, mentioned in a Jul 05 2016 comment of A010054. (End)
%F A274621 a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(5/2)*n^(5/4)). - _Vaclav Kotesovec_, Jul 05 2016
%t A274621 nmax = 40; CoefficientList[Series[Product[(1 - x^(2*k-1))^2 / (1 - x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 05 2016 *)
%Y A274621 If the signs are deleted we get A273225.
%Y A274621 Cf. A008441, A010054, A106507.
%K A274621 sign,easy
%O A274621 0,2
%A A274621 _N. J. A. Sloane_, Jul 03 2016