A138741 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q.

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

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..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of q^(-1/2) * (theta_2(q)^2 + 3 * theta_2(q^3)^2) / 4 in powers of q.

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

Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -2, 3, -2, 2, -4, 3, -2, ...].

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

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1 + (-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 / (48 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132003.

a(6*n + 3) = a(6*n + 5) = 0.

a(n) = (-1)^n * A116604(n). a(2*n) = A008441(n).

a(6*n) = A002175(n). a(6*n + 1) = 3 * A008441(n). a(6*n + 2) = 2 * A121444(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Dec 28 2023

Example

G.f. = 1 + 3*x + 2*x^2 + x^4 + 2*x^6 + 6*x^7 + 2*x^8 + 3*x^12 + 3*x^13 + ...
G.f. = q + 3*q^3 + 2*q^5 + q^9 + 2*q^13 + 6*q^15 + 2*q^17 + 3*q^25 + 3*q^27 + ...

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ x^(-1/2) (EllipticTheta[ 2, 0, x]^2 + 3 EllipticTheta[ 2, 0, x^3]^2) / 4, {x, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, # == 3, 2 - (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger[2 n + 1])]; (* Michael Somos, Sep 08 2015 *)
QP = QPochhammer; s = QP[q^2]^7*QP[q^3]*QP[q^12]^2 / (QP[q]^3*QP[q^4]^2* QP[q^6]^3) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

Prog

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

Crossrefs

Cf. A002175, A008441, A019669, A116604, A121444, A132003.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A285700 A290693 A131290 * A116604 A303913 A117406

Adjacent sequences: A138738 A138739 A138740 * A138742 A138743 A138744

Keyword

nonn,easy

Author

Michael Somos, Mar 27 2008

Status

approved