A263577 Expansion of psi(-x^2) * psi(x^3)^2 / f(-x^24) in powers of x where psi(), f() are Ramanujan theta functions.

1, 0, -1, 2, 0, -2, 0, 0, -1, 0, 0, -2, 2, 0, -2, 0, 0, 0, 2, 0, -2, 4, 0, 0, 0, 0, 0, 2, 0, -2, 4, 0, -1, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, 0, 2, 0, -3, 0, 0, -2, 0, 0, -2, 0, 0, -2, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, -4, 0, 0, -2

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(-x^12) * psi(-x^2) * chi(x^3)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of eta(q^2) * eta(q^6)^4 * eta(q^8) / (eta(q^3)^2 * eta(q^4) * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ 0, -1, 2, 0, 0, -3, 0, -1, 2, -1, 0, -2, 0, -1, 2, -1, 0, -3, 0, 0, 2, -1, 0, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 6^(3/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263571.

a(3*n) = A046113(n). a(3*n + 1) = 0. a(3*n + 2) = - A263548(n).

Example

G.f. = 1 - x^2 + 2*x^3 - 2*x^5 - x^8 - 2*x^11 + 2*x^12 - 2*x^14 + 2*x^18 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 2, 0, x^(3/2)]^2 / (2^(5/2) x QPochhammer[ x^24]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^12] EllipticTheta[ 2, Pi/4, x] / (2^(1/2) x^(1/4) QPochhammer[ x^3, -x^3]^2), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^4 * eta(x^8 + A) / (eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A)), n))};

Crossrefs

Cf. A046113, A263548, A263571.

Sequence in context: A360071 A358724 A325135 * A343631 A342561 A342562

Adjacent sequences: A263574 A263575 A263576 * A263578 A263579 A263580

Keyword

sign

Author

Michael Somos, Oct 21 2015

Status

approved