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.

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

Antti Karttunen, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

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

Cf. A033762, A262726, A262774.

Sequence in context: A246962 A112608 A058677 * A033762 A129449 A033798

Adjacent sequences: A262777 A262778 A262779 * A262781 A262782 A262783

Keyword

sign

Author

Michael Somos, Oct 01 2015

Status

approved