A212885 Expansion of phi(q) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.

1, -2, -4, 8, 6, -8, -8, 0, 12, -10, -8, 24, 8, -8, -16, 0, 6, -16, -12, 24, 24, -16, -8, 0, 24, -10, -24, 32, 0, -24, -16, 0, 12, -16, -16, 48, 30, -8, -24, 0, 24, -32, -16, 24, 24, -24, -16, 0, 8, -18, -28, 48, 24, -24, -32, 0, 48, -16, -8, 72, 0, -24, -32

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(-x) * phi(-x^2)^2 = phi(-x^2)^4 / phi(x) in powers of x where phi() is a Ramanujan theta function.

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045828.

G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 - x^k)^2 / (1 - x^(4*k))^2.

a(4*n) = A005875(n). a(4*n + 1) = -2 * A045834(n). a(4*n + 2) = - A005877(n) = -4 * A045828(n).

a(8*n) = A004015(n). a(8*n + 3) = A005878(n) = 8 * A008443(n). a(8*n + 4)= A005887(n). a(8*n + 5) = -2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.

Example

G.f. = 1 - 2*q - 4*q^2 + 8*q^3 + 6*q^4 - 8*q^5 - 8*q^6 + 12*q^8 - 10*q^9 + ...

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q]^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 30 2017 *)

Prog

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

Crossrefs

Cf. A004015, A004024, A005875, A005877, A005878, A005887, A008443, A045828, A045834, A213624.

Sequence in context: A247576 A339133 A246821 * A319078 A246631 A320153

Adjacent sequences: A212882 A212883 A212884 * A212886 A212887 A212888

Keyword

sign

Author

Michael Somos, May 29 2012

Status

approved