A246752 -id:A246752 - cp's OEIS frontend

A246838 Expansion of f(-x^2) * f(-x^12)^2 / f(x^1, x^5) in powers of x where f() is Ramanujan theta function.

1, -1, 0, 0, -1, 0, 1, -1, 0, 2, -1, 0, 0, 0, 0, 2, -1, 0, 1, -1, 0, 0, -2, 0, 0, -1, 0, 2, 0, 0, 0, -1, 0, 0, -1, 0, 3, -1, 0, 0, -1, 0, 2, -1, 0, 2, 0, 0, 0, -1, 0, 0, -1, 0, 2, -1, 0, 0, 0, 0, 1, -2, 0, 0, -2, 0, 0, -1, 0, 2, -1, 0, 2, 0, 0, 0, -1, 0, 0, 0

Offset: 0

Michael Somos, Sep 04 2014

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x - x^4 + x^6 - x^7 + 2*x^9 - x^10 + 2*x^15 - x^16 + x^18
+ ...
G.f. = q^3 - q^7 - q^19 + q^27 - q^31 + 2*q^39 - q^43 + 2*q^63 - q^67 + ...

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

Cf. A112604, A121361, A246752.

a[ n_] := SeriesCoefficient[ x^(1/4) EllipticTheta[ 2, Pi/4, x^(1/2)] QPochhammer[ x^12]^2 / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Aug 27 2015 *)

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

Expansion of q^(-3/4) * eta(q) * eta(q^4) * eta(q^6) * eta(q^12) / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 27^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246752.
a(3*n) = A112604(n). a(3*n + 1) = - A121361(n). a(3*n + 2) = 0.

A164272 Expansion of phi(q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Michael Somos, Aug 11 2009

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			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 + ...

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

Cf. A112604, A112605, A115978, A129449, A164273, A246752.

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 *)

{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))};

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

cp's OEIS Frontend

A246838 Expansion of f(-x^2) * f(-x^12)^2 / f(x^1, x^5) in powers of x where f() is Ramanujan theta function.

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