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A164272
Expansion of phi(q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
3
1, 2, 0, -2, -2, 0, 0, -4, 0, 2, 0, 0, -2, 4, 0, 0, 6, 0, 0, -4, 0, 4, 0, 0, 0, 2, 0, -2, -4, 0, 0, -4, 0, 0, 0, 0, -2, 4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, 6, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, 4, 0, -2, -4, 0, 0, -4, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 0, -8, 0, 4, 0, 0, 0, 4, 0, 0, -2, 0, 0, -4, 0
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2)^5 * eta(q^3)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 0, -1, 2, -4, 2, -1, 0, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112604.
a(n) = (-1)^n * A164273(n).
a(2*n + 1) = 2 * A129449(n). a(4*n) = A115978(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 2) = 0. a(4*n + 3) = -2 * A112605(n).
a(3*n) = A164273(n). a(3*n + 1) = 2 * A246752(n). a(3*n + 2) = 0. - Michael Somos, Sep 02 2015
EXAMPLE
G.f. = 1 + 2*q - 2*q^3 - 2*q^4 - 4*q^7 + 2*q^9 - 2*q^12 + 4*q^13 + 6*q^16 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^3], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164272[n_] := SeriesCoefficient[f[q, q]*f[-q^3, -q^3], {q, 0, n}]; Table[A164272[n], {n, 0, 50}] (* G. C. Greubel, Sep 16 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 11 2009
STATUS
approved