A246836 -id:A246836 - cp's OEIS frontend

A127786 Expansion of phi(q) * phi(q^2) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

1, 2, 2, 4, 0, -4, 0, -8, -2, 6, -8, 4, 0, -12, 0, -8, -4, 8, 10, 12, 0, -8, 0, -8, 8, 14, -8, 16, 0, -4, 0, -16, 6, 16, 16, 8, 0, -20, 0, -8, -8, 8, -16, 20, 0, -20, 0, -16, -8, 18, 10, 8, 0, -12, 0, -24, 0, 16, -24, 12, 0, -20, 0, -24, 12, 8, 16, 28, 0, -16, 0, -8, -10, 32, -8, 20, 0, -16, 0, -16, -8, 18, 32, 20, 0, -24, 0

Offset: 0

Michael Somos, Jan 29 2007

sign

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

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 - 4*q^5 - 8*q^7 - 2*q^8 + 6*q^9 - 8*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033763, A080963, A116597, A212885, A213622, A246832, A246833, A246835, A246836, A246837, A246954, A257873.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 08 2014 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)^3), n))};

Expansion of eta(q^2)^3 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^3) in powers of q.
Euler transform of period 8 sequence [ 2, -1, 2, -6, 2, -1, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 128 * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A213622. - Michael Somos, Sep 08 2014
a(8*n + 4) = a(8*n + 6) = 0.
a(n) = A080963(2*n). a(2*n) = A116597(n). a(2*n + 1) = 2 * A246836(n). a(4*n + 1) = 2 * A246835(n). a(4*n + 3) = 4 * A246833(n). - Michael Somos, Sep 08 2014
a(8*n) = A212885(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A246954(n). a(8*n + 3) = 4 * A246832(n). a(8*n + 5) = - 4 * A246837(n). a(8*n + 7) = - 8 * A033763(n). - Michael Somos, Sep 08 2014
a(3*n + 2) = 2 * A257873(n). - Michael Somos, May 11 2015

A246835 Expansion of psi(-x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -6, 4, -4, 7, -2, 8, -10, 4, -10, 9, -6, 8, -10, 4, -8, 16, -8, 9, -12, 8, -12, 20, -6, 8, -10, 8, -18, 11, -12, 8, -20, 12, -8, 20, -6, 20, -26, 8, -8, 15, -10, 16, -18, 12, -16, 20, -10, 16, -16, 8, -24, 24, -8, 21, -26, 8, -20, 20, -14, 8, -28

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 - 2*x + 3*x^2 - 6*x^3 + 4*x^4 - 4*x^5 + 7*x^6 - 2*x^7 + 8*x^8 + ...
G.f. = q - 2*q^5 + 3*q^9 - 6*q^13 + 4*q^17 - 4*q^21 + 7*q^25 - 2*q^29 + ...

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. A132969, A213622, A213625, A246836.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q^2]* EllipticTheta[2, 0, I*q^(1/2)]^2/(4*(-q)^(1/4)), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 29 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^4 * eta(x^8 + A)^2), n))};

Expansion of q^(-1/4) * eta(q)^2 * eta(q^4)^7 / (eta(q^2)^4 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 2, -2, -5, -2, 2, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 16 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246836.
a(n) = (-1)^n * A213625(n). a(2*n) = A213622(n). a(2*n + 1) = -2 * A132969(n).

cp's OEIS Frontend

A127786 Expansion of phi(q) * phi(q^2) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

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