A207541 Expansion of phi(q)^3 * phi(-q) in powers of q where phi() is a Ramanujan theta function.

1, 4, 0, -16, -8, 24, 0, -32, 24, 52, 0, -48, -32, 56, 0, -96, 24, 72, 0, -80, -48, 128, 0, -96, 96, 124, 0, -160, -64, 120, 0, -128, 24, 192, 0, -192, -104, 152, 0, -224, 144, 168, 0, -176, -96, 312, 0, -192, 96, 228, 0, -288, -112, 216, 0, -288, 192, 320, 0

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(-q^4)^4 + 4 * q * psi(-q^2)^4 = phi(q)^3 * phi(-q) = phi(q)^2 * phi(-q^2)^2 = psi(q)^4 * chi(-q^2)^6 = phi(-q^2)^6 / phi(-q)^2 = f(q)^6 / psi(q)^2 = f(q)^4 * chi(-q^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.

Euler transform of period 4 sequence [ 4, -10, 4, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A112610.

G.f.: Product_{k>0} (1 - x^(2*k))^14 / ((1 - x^k)^4 * (1 - x^(4*k))^6).

a(3*n + 2) = 24 * A208435(n). a(4*n + 2) = 0. a(2*n + 1) = 4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 3) = -16 * A097723(n). Convolution square of A139093.

Example

1 + 4*q - 16*q^3 - 8*q^4 + 24*q^5 - 32*q^7 + 24*q^8 + 52*q^9 - 48*q^11 + ...

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, -q], {q, 0, n}]; Table[A207541[n], {n, 0, 50}] (* G. C. Greubel, Dec 16 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^2, n))}

Crossrefs

Cf. A096727, A097723, A112610, A121613, A139093, A208435.

Sequence in context: A321610 A167314 A208451 * A158802 A230280 A030212

Adjacent sequences: A207538 A207539 A207540 * A207542 A207543 A207544

Keyword

sign

Author

Michael Somos, Feb 26 2012

Status

approved