A137829 Expansion of psi(q^2) / f(-q)^2 in powers of q where psi(), f() are Ramanujan theta functions.

1, 2, 6, 12, 25, 46, 86, 148, 255, 420, 686, 1088, 1712, 2634, 4020, 6036, 8988, 13214, 19282, 27840, 39923, 56750, 80160, 112384, 156660, 216958, 298894, 409420, 558119, 756950, 1022090, 1373760, 1838932, 2451366, 3255480, 4306920, 5678104, 7459634, 9768386

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

Cf. A137828, A137830.

Cf. Half of A201078, which gives another application.

Sequence in context: A210068 A210633 A304710 * A262196 A261667 A116562

Adjacent sequences: A137826 A137827 A137828 * A137830 A137831 A137832

Keyword

nonn

Author

Michael Somos, Feb 12 2008

Status

approved