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A262780
Expansion of phi(-x^6) * psi(x^4) + x * phi(-x^2) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.
2
1, 1, 0, -2, 1, 0, -2, 0, 0, 2, -2, 0, 1, 1, 0, -2, 0, 0, -2, -2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, -2, -2, 0, 2, 0, 0, 2, 1, 0, -2, 1, 0, 0, 0, 0, 4, -2, 0, 2, 0, 0, -2, 0, 0, -2, -2, 0, 0, -2, 0, 1, 0, 0, -2, 2, 0, -4, 0, 0, 2, 0, 0, 0, 3, 0, -2, 0, 0, -2, 0, 0
OFFSET
0,4
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 19 (mod 24), b(p^e) = (-1)^e * (e+1) if p == 7, 13 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
a(2*n) = A262774(n). a(2*n + 1) = A262726(n).
abs(a(n)) = A033762(n).
EXAMPLE
G.f. = 1 + x - 2*x^3 + x^4 - 2*x^6 + 2*x^9 - 2*x^10 + x^12 + x^13 + ...
G.f. = q + q^3 - 2*q^7 + q^9 - 2*q^13 + 2*q^19 - 2*q^21 + q^25 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, {1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, 1}[[Mod[n, 12, 1]]] DivisorSum[ 2 n + 1, KroneckerSymbol[ -3, #] &]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^6] EllipticTheta[ 2, 0, x^2] + EllipticTheta[ 4, 0, x^2] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2 + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (2 n + 1))];
PROG
(PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor(2*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 01 2015
STATUS
approved