A138745 Expansion of eta(q) * eta(q^3) * eta(q^4)^3 / (eta(q^2)^2 * eta(q^12)) in powers of q.

1, -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

0,4

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 = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (theta_4(q)^2 + 3 * theta_4(q^3)^2) / 4 in powers of q.

Expansion of psi(-q) * psi(q^2) * chi(-q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

Euler transform of period 12 sequence [ -1, 1, -2, -2, -1, 0, -1, -2, -2, 1, -1, -2, ...].

Moebius transform is period 24 sequence [ -1, 2, -2, 0, -1, 4, 1, 0, 2, 2, 1, 0, -1, -2, -2, 0, -1, -4, 1, 0, 2, -2, 1, 0, ...].

a(n) = -b(n) where b() is multiplicative with b(2^e) = -1 if e>0, b(3^e) = 2 - (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A125079.

G.f.: 1 + Sum_{k>0} (-1)^k * ( f(6*k - 1) + 2 * f(6*k - 3) + f(6*k - 5) ) where f(k) := x^k / (1 + x^k).

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

a(n) = - A138746(n) unless n=0. a(n) = (-1)^n * A125061(n).

a(2*n) = A125061(n). a(2*n + 1) = - A138741(n).

Example

G.f. = 1 - q + q^2 - 3*q^3 + q^4 - 2*q^5 + 3*q^6 + q^8 - q^9 + 2*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q]^2 + 3 EllipticTheta[ 4, 0, q^3]^2) / 4, {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], - Times @@ (Which[ # < 3, -(-1)^#, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* Michael Somos, Sep 08 2015 *)
QP = QPochhammer; s = QP[q]*QP[q^3]*(QP[q^4]^3/(QP[q^2]^2*QP[q^12])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); - prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2 )))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^3 / (eta(x^2 + A)^2 * eta(x^12 + A)), n))};

Crossrefs

Cf. A125061, A125079, A138741, A138746.

Sequence in context: A016469 A274342 A138746 * A138952 A138950 A125061

Adjacent sequences: A138742 A138743 A138744 * A138746 A138747 A138748

Keyword

sign

Author

Michael Somos, Mar 27 2008

Status

approved