A246954 -id:A246954 - 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

A246953 Expansion of phi(-x) * psi(x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 2, -4, 3, -2, 6, -4, 4, -6, 4, -4, 7, -8, 2, -8, 8, -4, 10, -4, 4, -10, 10, -8, 9, -4, 6, -12, 8, -6, 10, -12, 4, -14, 8, -4, 16, -10, 8, -8, 9, -10, 12, -12, 8, -12, 12, -4, 20, -10, 6, -20, 8, -6, 10, -12, 8, -20, 18, -8, 11, -12, 12, -16, 8, -6, 20

Offset: 0

Michael Somos, Sep 08 2014

sign

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

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

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. A045828, A213624, A213625, A246954.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x]^2/(4 x^(1/2)), {x, 0, n}];

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

Expansion of psi(x^2) * psi(-x)^2 = psi(-x)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^4 / eta(q^2)^3 in powers of q.
Euler transform of period 4 sequence [ -2, 1, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 128^(1/2) * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A246954.
G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^k) * (1 + x^(2*k))^4.
a(n) = (-1)^n * A045828(n). a(2*n) = A213625(n). a(2*n + 1) = - 2 * A213624(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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula