A208605 Expansion of q * psi(q^8) / phi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

1, -2, 4, -8, 14, -24, 40, -64, 101, -156, 236, -352, 518, -752, 1080, -1536, 2162, -3018, 4180, -5744, 7840, -10632, 14328, -19200, 25591, -33932, 44776, -58816, 76918, -100176, 129952, -167936, 216240, -277476, 354864, -452392, 574958, -728568, 920600

Offset

1,2

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^16)^2 / (eta(q^2)^5 * eta(q^8)) in powers of q.

Euler transform of period 16 sequence [ -2, 3, -2, 1, -2, 3, -2, 2, -2, 3, -2, 1, -2, 3, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/4 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A208603.

a(n) = -(-1)^n * A123655(n). a(2*n) = -2 * A107035(n). a(2*n + 1) = A093160(n). Convolution inverse of A208603.

Example

q - 2*q^2 + 4*q^3 - 8*q^4 + 14*q^5 - 24*q^6 + 40*q^7 - 64*q^8 + 101*q^9 + ...

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^2* eta[q^4]^2*eta[q^16]^2/(eta[q^2]^5*eta[q^8]), {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jan 23 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2 / (eta(x^2 + A)^5 * eta(x^8 + A)), n))}

Crossrefs

Cf. A093160, A107035, A123655, A208603.

Sequence in context: A069253 A004402 A015128 * A123655 A084683 A271493

Adjacent sequences: A208602 A208603 A208604 * A208606 A208607 A208608

Keyword

sign

Author

Michael Somos, Feb 29 2012

Status

approved