A138952 Expansion of (eta(q^2)^7 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^3 * eta(q^6)^3) - 1) / 2 in powers of q.

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

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 (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.

a(n) = -(-1)^n * A138950(n). 2 * a(n) = A138951(n).

a(2*n) = - A138950(n). a(2*n + 1) = A116604(n). - Michael Somos, Sep 07 2015

a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). - Michael Somos, Sep 07 2015

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

Cf. A116604, A138950, A138951, A258277, A258278.

Sequence in context: A274342 A138746 A138745 * A138950 A125061 A163746

Adjacent sequences: A138949 A138950 A138951 * A138953 A138954 A138955

Keyword

sign,mult

Author

Michael Somos, Apr 03 2008

Status

approved