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
Euler transform of period 18 sequence [ -2, -1, 0, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, 0, -1, -2, -2, ...].
a(n) = b(4n+1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (e+1)(-1)^e if p == 5 (mod 12).
G.f.: Product_{k>0} (1+x^(3n))^2(1-x^n)(1-x^(9n))/((1+x^n)(1+x^9n)).
a(3n+2) = 0.
Expansion of phi(-q)*phi(-q^9)/chi(-q^3)^2 in powers of q where phi(),chi() are Ramanujan theta functions.
MATHEMATICA
QP = QPochhammer; s = (QP[q]*QP[q^6]*(QP[q^9]/QP[q^3]))^2/QP[q^2]/QP[q^18]+ O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
PROG
(PARI) {a(n)=if(n<0, 0, n=4*n+1; dirmul(vector(n, k, kronecker(12, k)), vector(n, k, kronecker(-12, k)))[n])}
(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=4*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%4==3, (1+(-1)^e)/2, (e+1)*if(p%3==2, (-1)^e, 1)))))) }
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^6+A)*eta(x^9+A)/ eta(x^3+A))^2/ eta(x^2+A)/ eta(x^18+A), n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 22 2006
STATUS
approved