OFFSET
0,2
COMMENTS
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.
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
KEYWORD
sign
AUTHOR
Michael Somos, Feb 18 2006, Apr 03 2008
STATUS
approved