A134746 Expansion of 1+k in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.

1, 4, 0, -16, 0, 56, 0, -160, 0, 404, 0, -944, 0, 2072, 0, -4320, 0, 8648, 0, -16720, 0, 31360, 0, -57312, 0, 102364, 0, -179104, 0, 307672, 0, -519808, 0, 864960, 0, -1419456, 0, 2299832, 0, -3682400, 0, 5831784, 0, -9141808, 0, 14194200, 0, -21842368, 0

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (phi(q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

Expansion of (eta(q^8) / eta(q))^4 * (eta(q^2) / eta(q^4))^14 in powers of q.

Euler transform of period 8 sequence [ 4, -10, 4, 4, 4, -10, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A210066.

G.f.: ( (Sum_{k in Z} x^(k^2)) / (Sum_{k in Z} x^(2*k^2)) )^2 = ( Product_{k>0} (1 + x^k)^2 * (1 + x^(4*k))^2 / (1 + x^(2*k))^5 )^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u * (2 - u) * v^2.

a(2*n) = 0 unless n=0. a(2*n + 1) = 4 * A001938(n) = A127393(n).

a(n) = (-1)^n * A210067(n). Convolution inverse of A210066. - Michael Somos, Oct 16 2015

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 34 + 24*sqrt(2) - 4*sqrt(140 + 99*sqrt(2)). - Simon Plouffe, Mar 04 2021

Example

G.f. = 1 + 4*q - 16*q^3 + 56*q^5 - 160*q^7 + 404*q^9 - 944*q^11 + 2072*q^13 + ...

Mathematica

CoefficientList[Series[(QPochhammer[x^8]/QPochhammer[x])^4 (QPochhammer[x^2]/QPochhammer[x^4])^14, {x, 0, 50}], x] (* Jan Mangaldan, Mar 21 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)

Prog

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

Crossrefs

Cf. A001938, A127393, A210066, A210067.

Sequence in context: A170771 A170772 A079986 * A210067 A199572 A003195

Adjacent sequences: A134743 A134744 A134745 * A134747 A134748 A134749

Keyword

sign

Author

Michael Somos, Nov 07 2007

Status

approved