OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 0, -1, -1, -2, 0, -1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e * (e+1) if p == 7 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A033762(n). a(2*n) = A112604(n). a(2*n + 1) = -A112605(n). a(3*n) = A129451(n). a(3*n + 1) = -a(n). a(3*n + 2) = 0.
EXAMPLE
G.f. = 1 - x - 2*x^3 + x^4 + 2*x^6 - 2*x^9 + 2*x^10 + x^12 - x^13 - 2*x^15 + ...
G.f. = q - q^3 - 2*q^7 + q^9 + 2*q^13 - 2*q^19 + 2*q^21 + q^25 - q^27 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -4, m/d], {d, Divisors[ m]}]]]; (* Michael Somos, Jul 09 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, d) * kronecker( 12, n/d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)/ (eta(x^2 + A) * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Apr 16 2007
STATUS
approved