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A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a Ramanujan theta function. 5
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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(exp(-Pi)^(n-1)*a(n),n=1..infinity) = (-2+2*3^(1/2))^(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

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.

Sequence in context: A124449 A262877 A141094 * A113920 A081848 A079988

Adjacent sequences:  A132969 A132970 A132971 * A132973 A132974 A132975

KEYWORD

nonn,changed

AUTHOR

Michael Somos, Sep 06 2007

STATUS

approved

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Last modified November 16 20:09 EST 2019. Contains 329204 sequences. (Running on oeis4.)