A263444 Expansion of psi(q)^2 * chi(-q^6)^2 * f(-q^6) in powers of q where psi(), chi(), f() are Ramanujan theta functions.

1, 2, 1, 2, 2, 0, 0, -4, -3, -4, -4, 2, -6, 0, 2, 0, 2, -6, 4, -4, 0, 0, -8, 4, 0, 14, 2, 2, 12, 0, 8, -4, -3, 0, -4, 4, -4, 0, 4, 8, 12, -6, 0, -12, -6, 0, -8, 4, -6, 14, 5, 0, 0, 0, 0, -8, -6, -24, -12, 2, 0, 0, 6, 8, 2, -12, 8, -12, -6, 0, -8, 4, -12, 24, 6

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

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 15552^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263433.

a(8*n + 5) = a(9*n + 6) = 0.

Example

G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 - 4*x^7 - 3*x^8 - 4*x^9 - 4*x^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^2 QPochhammer[ q^6, q^12]^2 QPochhammer[ q^6] / (4 q^(1/4)), {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^3 / (eta(x + A)^2 * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A263433.

Sequence in context: A230322 A174950 A159906 * A276321 A152196 A024375

Adjacent sequences: A263441 A263442 A263443 * A263445 A263446 A263447

Keyword

sign

Author

Michael Somos, Oct 18 2015

Status

approved