A190611 Expansion of f(q^3) * f(-q^8) * chi(-q^12) / chi(q) in powers of q where f(), chi() are Ramanujan theta functions.

1, -1, 1, -1, 1, -2, 1, -2, 1, -1, 2, -2, 1, 0, 2, -2, 1, 0, 1, 0, 2, -2, 2, 0, 1, -3, 0, -1, 2, -2, 2, -2, 1, -2, 0, -4, 1, 0, 0, 0, 2, 0, 2, 0, 2, -2, 0, 0, 1, -3, 3, 0, 0, -2, 1, -4, 2, 0, 2, -2, 2, 0, 2, -2, 1, 0, 2, 0, 0, 0, 4, 0, 1, -2, 0, -3, 0, -4, 0

Offset

0,6

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 = 0..5000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q) * eta(q^4) * eta(q^6)^3 * eta(q^8) / (eta(q^2)^2 * eta(q^3) * eta(q^24)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 96^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129402.

a(n) = (-1)^n * A000377(n). a(24*n + 13) = a(24*n + 17) = a(24*n + 19) = a(24*n + 23) = 0.

a(2*n) = A000377(n). a(2*n + 1) = - A129402(n). a(3*n) = a(n). - Michael Somos, Nov 11 2015

Example

G.f. = 1 - q + q^2 - q^3 + q^4 - 2*q^5 + q^6 - 2*q^7 + q^8 - q^9 + 2*q^10 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n DivisorSum[ n, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q] QPochhammer[ -q^3] QPochhammer[ q^8] QPochhammer[ q^12, -q^12], {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv( n, d, kronecker( -6, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 * eta(x^8 + A) / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^24 + A)), n))};

Crossrefs

Cf. A000377, A129402.

Sequence in context: A261122 A115660 A128581 * A000377 A192013 A329618

Adjacent sequences: A190608 A190609 A190610 * A190612 A190613 A190614

Keyword

sign

Author

Michael Somos, May 14 2011

Status

approved