A246926 Expansion of phi(x)^2 * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, 5, 8, 4, 4, 13, 12, 4, 5, 16, 24, 8, 4, 20, 12, 8, 9, 20, 32, 4, 12, 29, 12, 8, 8, 36, 40, 8, 8, 20, 24, 16, 8, 25, 40, 12, 12, 32, 24, 12, 13, 48, 40, 8, 8, 40, 36, 8, 16, 20, 56, 16, 12, 52, 12, 20, 13, 36, 56, 16, 20, 40, 24, 8, 8, 45, 72, 12, 16, 52

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/3) * eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^5 * eta(q^4)^5 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [5, -7, 4, -2, 5, -7, 5, -2, 4, -7, 5, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 72^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246927.

2 * a(n) = A246928(3*n + 1).

Example

G.f. = 1 + 5*x + 8*x^2 + 4*x^3 + 4*x^4 + 13*x^5 + 12*x^6 + 4*x^7 + 5*x^8 + ...
G.f. = q + 5*q^4 + 8*q^7 + 4*q^10 + 4*q^13 + 13*q^16 + 12*q^19 + 4*q^22 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}]; (* Michael Somos, Jan 08 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^5 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma0(36), 3/2), 210); A[2] + 5*A[5];

Crossrefs

Cf. A246927, A246928.

Sequence in context: A366072 A019649 A224833 * A199288 A099878 A167901

Adjacent sequences: A246923 A246924 A246925 * A246927 A246928 A246929

Keyword

nonn

Author

Michael Somos, Sep 07 2014

Status

approved