A208274 - OEIS

A208274

Expansion of phi(q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 0, 0, -4, 0, 0, 0, 10, 0, 0, 0, -20, 0, 0, 0, 36, 0, 0, 0, -64, 0, 0, 0, 110, 0, 0, 0, -180, 0, 0, 0, 288, 0, 0, 0, -452, 0, 0, 0, 692, 0, 0, 0, -1044, 0, 0, 0, 1554, 0, 0, 0, -2276, 0, 0, 0, 3296, 0, 0, 0, -4724, 0, 0, 0, 6696, 0, 0, 0, -9408, 0, 0

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OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Differs from A127391 only at n=0. - R. J. Mathar, Mar 18 2012

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 eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5) in powers of q.

Euler transform of period 16 sequence [ 2, -3, 2, -3, 2, -3, 2, 2, 2, -3, 2, -3, 2, -3, 2, 0, ...].

G.f. A(x) satisfies A(x)^2 - 2*A(x) + 2 = A134746(x^2), which means (phi(q) / phi(q^4) - 1)^2 + 1 = (phi(q^2) / phi(q^4))^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - 2*u + 2) * (v^2 - 2*v + 2) - v^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = 4 * u * (u - 1) * (2 - u) * v * (v - 1) * (2 - v) - (u - v)^4.

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = g(t) where q = exp(2 Pi i t) and g() is g.f. for A112128.

a(4*n) = 0 unless n=0. a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A079006(n). a(n) = (-1)^n * A208604(n). Convolution inverse is A112128.

EXAMPLE

1 + 2*q - 4*q^5 + 10*q^9 - 20*q^13 + 36*q^17 - 64*q^21 + 110*q^25 - 180*q^29 + ...

MATHEMATICA

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^5), n))}

CROSSREFS

Cf. A079006, A112128, A134746, A208604.

Sequence in context: A134309 A051516 A236799 * A208604 A127391 A262162

Adjacent sequences: A208271 A208272 A208273 * A208275 A208276 A208277

KEYWORD

sign

AUTHOR

Michael Somos, Mar 12 2012

STATUS

approved