A147699 Expansion of f(x) * f(x^5) / phi(-x^10)^2 in powers of x where f(), phi() are Ramanujan theta functions.

1, 1, -1, 0, 0, 0, 1, -2, 0, 0, 2, 3, -5, 0, 0, 2, 4, -8, 0, 0, 5, 8, -14, 0, 0, 6, 12, -22, 0, 0, 13, 21, -36, 0, 0, 16, 30, -54, 0, 0, 28, 48, -83, 0, 0, 38, 68, -120, 0, 0, 60, 102, -176, 0, 0, 80, 143, -250, 0, 0, 122, 207, -356, 0, 0, 162, 284, -494, 0, 0

Offset

0,8

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^(-1/4) * eta(q^2)^3 * eta(q^20) / (eta(q) * eta(q^4) * eta(q^5) * eta(q^10)) in powers of q.

Euler transform of period 20 sequence [ 1, -2, 1, -1, 2, -2, 1, -1, 1, 0, 1, -1, 1, -2, 2, -1, 1, -2, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (5/4)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147702.

a(5*n + 2) = a(5*n + 3) = 0.

a(n) = A138532(2*n + 1). a(5*n + 1) = A145722(n).

a(5*n) = A261866(n). a(5*n + 2) = - A261526(n). - Michael Somos, Sep 03 2015

Example

G.f. = 1 + x - x^2 + x^6 - 2*x^7 + 2*x^10 + 3*x^11 - 5*x^12 + 2*x^15 + ...
G.f. = q + q^5 - q^9 + q^25 - 2*q^29 + 2*q^41 + 3*q^45 - 5*q^49 + 2*q^61 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^10]^2, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^10 + A)), n))};

Crossrefs

Cf. A138532, A145722, A147702, A261526, A261866.

Sequence in context: A327950 A364806 A024870 * A250023 A151669 A115509

Adjacent sequences: A147696 A147697 A147698 * A147700 A147701 A147702

Keyword

sign

Author

Michael Somos, Nov 10 2008

Status

approved