OFFSET
1,7
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless e=0, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
EXAMPLE
G.f. = q - q^2 + q^3 - q^4 - q^6 + 2*q^7 - q^8 + q^9 - q^12 + 2*q^13 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[n, KroneckerSymbol[ -12, #] &]]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ (4 + EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]) / 6, {q, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, May 07 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker(-12, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A))) / 3, n))}; /* Michael Somos, May 06 2015 */
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Jan 29 2008
STATUS
approved