A261877 Expansion of psi(x^4) / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 0, 0, 2, 1, 0, 4, 2, 0, 8, 4, 0, 15, 8, 0, 26, 14, 0, 44, 24, 0, 72, 40, 0, 115, 64, 0, 180, 100, 0, 276, 154, 0, 416, 232, 0, 618, 344, 0, 906, 505, 0, 1312, 730, 0, 1880, 1044, 0, 2666, 1480, 0, 3746, 2076, 0, 5220, 2888, 0, 7216, 3988, 0, 9903, 5464, 0

Offset

0,4

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^(-1/2) * eta(q^6) * eta(q^8)^2 / (eta(q^3)^2 * eta(q^4)) in powers of q.

Euler transform of period 24 sequence [ 0, 0, 2, 1, 0, 1, 0, -1, 2, 0, 0, 2, 0, 0, 2, -1, 0, 1, 0, 1, 2, 0, 0, 0, ...].

2 * a(n) = A143068(2*n + 1). a(3*n + 2) = 0.

Convolution inverse is A262929. - Michael Somos, Oct 22 2017

Example

G.f. = 1 + 2*x^3 + x^4 + 4*x^6 + 2*x^7 + 8*x^9 + 4*x^10 + 15*x^12 + ...
G.f. = q + 2*q^7 + q^9 + 4*q^13 + 2*q^15 + 8*q^19 + 4*q^21 + 15*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/2 (x^2)^(-1/4) EllipticTheta[ 2, 0, x^2] / EllipticTheta[ 4, 0, x^3], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) * eta(x^8 + A)^2 / (eta(x^3 + A)^2 * eta(x^4 + A)), n))};

Crossrefs

Cf. A143068, A262929.

Sequence in context: A122792 A348218 A138002 * A062296 A249343 A369206

Adjacent sequences: A261874 A261875 A261876 * A261878 A261879 A261880

Keyword

nonn

Author

Michael Somos, Sep 09 2015

Status

approved