A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a Ramanujan theta function.

1, 3, 3, 3, 6, 9, 12, 15, 21, 30, 36, 45, 60, 78, 96, 117, 150, 189, 228, 276, 342, 420, 504, 603, 732, 885, 1050, 1245, 1488, 1773, 2088, 2454, 2901, 3420, 3996, 4662, 5460, 6378, 7404, 8583, 9972, 11565, 13344, 15378, 17748, 20448, 23472, 26910, 30876

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

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 12 sequence [ 3, -3, 2, 0, 3, -2, 3, 0, 2, -3, 3, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 + u*v) * (u*v - 1)^3 - (u - u^4) * (v - v^4).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (4 - 2*u + u^2) - v^3 * (1 + u + u^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (2 + u1 * u2) - u3 * u6 * (1 + u1 + u2).

G.f. is a period 1 Fourier series which satisfies f(-1/(144*t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A062244.

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

a(n) = 3 * A132975(n) unless n=0.

Empirical: Sum_{n>=1} exp(-Pi)^(n-1)*a(n) = (-2 + 2*sqrt(3))^(1/3). - Simon Plouffe, Feb 20 2011

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - (-1)^(n+1)*x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - (-1)^n*x^(3*n + 2))) ). Cf. A273845. - Peter Bala, Dec 23 2021

Example

G.f. = 1 + 3*q + 3*q^2 + 3*q^3 + 6*q^4 + 9*q^5 + 12*q^6 + 15*q^7 + 21*q^8 + ...

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^3 / (1 + x^(6*k-3)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^3 / QPochhammer[ -q^3, q^6], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^2), n))};

Crossrefs

Cf. A062244, A132975, A273845.

Sequence in context: A262877 A348224 A141094 * A113920 A081848 A079988

Adjacent sequences: A132969 A132970 A132971 * A132973 A132974 A132975

Keyword

nonn

Author

Michael Somos, Sep 06 2007

Status

approved