OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Moreno (preprint) calls this |Phi_n|. - N. J. A. Sloane, Sep 01 2018
REFERENCES
Lusztig, G., Irreducible representation of finite classical groups, Inventiones Math., 43 (1977), 125-175. See p. 135.
Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table II.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Márton Balázs, Dan Fretwell, and Jessica Jay, Interacting Particle Systems and Jacobi style identities, arXiv:2011.05006 [math.PR], 2020.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/6) * eta(q^4)^2 / (eta(q)^2 * eta(q^2)) in powers of q.
Euler transform of period 4 sequence [ 2, 3, 2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 96^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137830.
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k).
2 * a(n) = A137828(4*n + 1).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 25*x^4 + 46*x^5 + 86*x^6 + 148*x^7 + ...
G.f. = q + 2*q^7 + 6*q^13 + 12*q^19 + 25*q^25 + 46*q^31 + 86*q^37 + 148*q^43 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]^2), {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k))^2 / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A)^2 * eta(x^2 + A)), n))};
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Michael Somos, Feb 12 2008
STATUS
approved