A138950 Expansion of (2 - 3 * phi(q^3)^2 + phi(q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

1, 1, -3, 1, 2, -3, 0, 1, 1, 2, 0, -3, 2, 0, -6, 1, 2, 1, 0, 2, 0, 0, 0, -3, 3, 2, -3, 0, 2, -6, 0, 1, 0, 2, 0, 1, 2, 0, -6, 2, 2, 0, 0, 0, 2, 0, 0, -3, 1, 3, -6, 2, 2, -3, 0, 0, 0, 2, 0, -6, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, -9, 0, 0, -6, 0, 2, 1, 2

Offset

1,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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 - eta(q)^2 * eta(q^2) * eta(q^6)^3 / (eta(q^3)^2 * eta(q^4) * eta(q^12))) / 2 in powers of q.

Moebius transform is period 12 sequence [ 1, 0, -4, 0, 1, 0, -1, 0, 4, 0, -1, 0, ...].

a(n) is multiplicative with a(2^e) = 1, 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).

G.f.: Sum_{k>0} f(3*k - 2) + f(3*k - 1) - 2 * f(3*k) where f(n) := x^n / (1 + x^(2*n)).

a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n). a(2*n + 1) = A116604(n).

-2 * a(n) = A138949(n) unless n=0. a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n).

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, DivisorSum[ n, KroneckerSymbol[ -4, n/#] {1, 1, -2}[[Mod[#, 3, 1]]] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ (2 - 3 EllipticTheta[ 3, 0, q^3]^2 + EllipticTheta[ 3, 0, q]^2) / 4, {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, - 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

Cf. A008441, A116604, A122856, A122865, A138949.

Sequence in context: A138746 A138745 A138952 * A125061 A163746 A004591

Adjacent sequences: A138947 A138948 A138949 * A138951 A138952 A138953

Keyword

sign,mult

Author

Michael Somos, Apr 03 2008

Status

approved