OFFSET
0,2
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
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
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 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
N. J. A. Sloane, Jul 03 2016
STATUS
approved