A274621 Coefficients in the expansion Product_{ n>=1 } (1-q^(2n-1))^2/(1-q^(2n))^2.

1, -2, 3, -6, 11, -18, 28, -44, 69, -104, 152, -222, 323, -460, 645, -902, 1254, -1722, 2343, -3174, 4278, -5722, 7601, -10056, 13250, -17358, 22623, -29382, 38021, -48984, 62857, -80404, 102528, -130282, 165002, -208398, 262495, -329666, 412878, -515840, 642941, -799362, 991478

Offset: 0

Author

N. J. A. Sloane, Jul 03 2016

Keywords

• sign
• easy

Comments

This is the reciprocal of the g.f. for A008441.

From Wolfdieter Lang, Jul 05 2016: (Start)

The g.f. is the square of the one for A106507.

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)

References

• 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.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
• R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
• R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts

Crossrefs

If the signs are deleted we get A273225.

Cf. A008441, A010054, A106507.

Programs

Mathematica

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 *)

Formula

From Wolfdieter Lang, Jul 05 2016: (Start)

G.f.: 1/(theta_2(0, sqrt(q))/(2*q^(1/8)))^2, with the Jacobi theta_2 function.

G.f.: 1/(Sum_{n >= 0} q^(n*(n+1)/2))^2.

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)

a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016