A116604 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q.

1, -3, 2, 0, 1, 0, 2, -6, 2, 0, 0, 0, 3, -3, 2, 0, 0, 0, 2, -6, 2, 0, 2, 0, 1, -6, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, -9, 0, 0, 1, 0, 4, -6, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, -6, 2, 0, 2, 0, 1, -6, 4, 0, 0, 0, 0, -6, 2, 0, 0, 0, 4, -3, 2, 0, 2, 0, 2, -6, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, -6, 4, 0, 0, 0, 2, -12, 2, 0, 0, 0, 4, 0, 0

Offset

0,2

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 = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^k + x^(2*k)) * (1 + x^(6*k)).

G.f.: Sum_{k>=0} x^(3*k) / (1 + x^(6*k + 1)) - 2*x^(3*k + 1) /(1 + x^(6*k+3)) + x^(3*k + 2) / (1 + x^(6*k + 5)).

Expansion of psi(q^2)^2 - 3 * q * psi(q^6)^2 in powers of q where psi() is a Ramanujan theta function.

Euler transform of period 12 sequence [ -3, -1, -2, -2, -3, 0, -3, -2, -2, -1, -3, -2, ...].

Moebius transform is period 24 sequence [ 1, -1, -4, 0, 1, 4, -1, 0, 4, -1, -1, 0, 1, 1, -4, 0, 1, -4, -1, 0, 4, 1, -1, 0, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = -1 + 2 * (-1)^e, b(p^e) = e + 1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A121450.

a(6*n + 3) = a(6*n + 5) = 0. a(6*n) = A002175(n). a(2*n) = A008441(n).

Example

1 - 3*x + 2*x^2 + x^4 + 2*x^6 - 6*x^7 + 2*x^8 + 3*x^12 - 3*x^13 + ...
q - 3*q^3 + 2*q^5 + q^9 + 2*q^13 - 6*q^15 + 2*q^17 + 3*q^25 - 3*q^27 + ...

Mathematica

QP = QPochhammer; s = QP[q]^3*QP[q^4]*(QP[q^12]/(QP[q^2]^2*QP[q^3])) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, n/d) * [ -2, 1, 1][d%3 + 1]))}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, 0, if( p==3, -1 + 2 * (-1)^e, if( p%12 < 6, e+1, (1 + (-1)^e) / 2)))))) }
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)), n))}

Crossrefs

Cf. A002175, A008441, A121450.

Sequence in context: A290693 A131290 A138741 * A303913 A117406 A290216

Adjacent sequences: A116601 A116602 A116603 * A116605 A116606 A116607

Keyword

sign

Author

Michael Somos, Feb 18 2006, Apr 03 2008

Status

approved