A213607 - OEIS

A213607

Expansion of psi(x^4) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 2, 0, 3, 2, 4, 0, 3, 3, 4, 0, 4, 3, 6, 0, 7, 3, 4, 0, 6, 5, 4, 0, 7, 4, 8, 0, 7, 5, 8, 0, 5, 4, 10, 0, 8, 5, 6, 0, 7, 7, 8, 0, 11, 5, 10, 0, 9, 8, 8, 0, 5, 4, 12, 0, 14, 5, 8, 0, 10, 8, 8, 0, 11, 8, 10, 0, 10, 9, 14, 0, 10, 5, 10, 0, 15, 7, 6, 0, 10, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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 q^(-5/6) * eta(q^3)^3 * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A213618.

a(4*n + 3) = 0. a(4*n + 2) = 2 * A213023(n).

EXAMPLE

1 + x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^6 + 3*x^8 + 3*x^9 + 4*x^10 + ...

q^5 + q^11 + 2*q^17 + 3*q^29 + 2*q^35 + 4*q^41 + 3*q^53 + 3*q^59 + 4*q^65 + ...

MATHEMATICA

QP := QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^3]^3*QP[q^8]^2 )/( QP[q]*QP[q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 07 2018 *)

PROG

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

CROSSREFS

Cf. A213023, A213618.

Sequence in context: A349134 A338101 A338490 * A298932 A089196 A208435

Adjacent sequences: A213604 A213605 A213606 * A213608 A213609 A213610

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 16 2012

STATUS

approved