A138951 Expansion of eta(q^2)^7 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^3 * eta(q^6)^3) in powers of q.

1, 2, -2, -6, -2, 4, 6, 0, -2, 2, -4, 0, 6, 4, 0, -12, -2, 4, -2, 0, -4, 0, 0, 0, 6, 6, -4, -6, 0, 4, 12, 0, -2, 0, -4, 0, -2, 4, 0, -12, -4, 4, 0, 0, 0, 4, 0, 0, 6, 2, -6, -12, -4, 4, 6, 0, 0, 0, -4, 0, 12, 4, 0, 0, -2, 8, 0, 0, -4, 0, 0, 0, -2, 4, -4, -18, 0

Offset

0,2

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

L.-C. Shen, On the Modular Equations of Degree 3, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1108, Eq. (3.24).

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

Euler transform of period 12 sequence [ 2, -5, 0, -2, 2, -4, 2, -2, 0, -5, 2, -2, ...].

Moebius transform is period 24 sequence [ 2, -4, -8, 0, 2, 16, -2, 0, 8, -4, -2, 0, 2, 4, -8, 0, 2, -16, -2, 0, 8, 4, -2, 0, ...].

a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = -1 if e>0, b(3^e) = -1 + 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)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129447.

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

G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 - x^(2*k) + x^(4*k)) / ((1 + x^(2*k))^2 * (1 - x^k + x^(2*k))^2).

a(n) = (-1)^n * A138949(n).

Example

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

Mathematica

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

Prog

(PARI) {a(n) = if( n<1, n==0, -2 * (-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, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, -1 + 2 * (-1)^e, if(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^2 + A)^7 * eta(x^3 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^6 + A)^3), n))};

Crossrefs

Cf. A138949.

Sequence in context: A062401 A286383 A138949 * A163370 A278159 A071796

Adjacent sequences: A138948 A138949 A138950 * A138952 A138953 A138954

Keyword

sign

Author

Michael Somos, Apr 03 2008

Status

approved