A145723 Expansion of q^(-1) * f(q) * chi(-q^5) / f(-q^20) in powers of q where f(), chi() are Ramanujan theta functions.

1, 1, -1, 0, 0, -2, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 2, 0, 0, 2, 2, 0, 0, 0, -4, -1, 0, 0, 0, 2, 0, -1, 0, 0, 0, -2, 2, 0, 0, 3, 4, -2, 0, 0, -8, -3, 0, 0, 0, 5, 0, -2, 0, 0, 0, -3, 4, 0, 0, 6, 8, -2, 0, 0, -14, -4, 0, 0, 0, 8, 0, -3, 0, 0, 0, -6, 8, 0, 0

Offset

-1,6

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145722.

a(4*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.

a(4*n) = A138527(n). a(4*n + 1) = - A147699(n).

Convolution inverse of A145724.

Example

G.f. = 1/q + 1 - q - 2*q^4 - q^5 + q^9 - q^15 + 2*q^16 + 2*q^19 + 2*q^20 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q]  QPochhammer[ q^5, q^10] / QPochhammer[ q^20], {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A138527, A145722, A145724, A147699.

Sequence in context: A352553 A077655 A117886 * A085977 A288424 A127325

Adjacent sequences: A145720 A145721 A145722 * A145724 A145725 A145726

Keyword

sign

Author

Michael Somos, Nov 06 2008

Status

approved