OFFSET
1,3
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 (phi(q) * phi(-q^2) * chi(-q^3) / chi(q^3) - 1) / 2 in powers of q where phi(), chi() are Ramanujan theta functions.
Moebius transform is period 24 sequence [1, -2, -4, 0, 1, 8, -1, 0, 4, -2, -1, 0, 1, 2, -4, 0, 1, -8, -1, 0, 4, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = -1 + 2 * (-1)^e, a(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
a(12*n + 7) = a(12*n + 11) = 0.
EXAMPLE
G.f. = q - q^2 - 3*q^3 - q^4 + 2*q^5 + 3*q^6 - q^8 + q^9 - 2*q^10 + 3*q^12 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -4, n/#] {1, 1, -2}[[Mod[#, 3, 1]]] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -1, # == 3, -1 + 2 (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2] QPochhammer[ q^3] / QPochhammer[ -q^3] - 1) / 2, {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker(-4, n/d) * [-2, 1, 1][d%3 + 1]))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, -1 + 2 * (-1)^e, p%12 < 6, e+1, 1-e%2 )))};
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Apr 03 2008
STATUS
approved