A194094 Expansion of (2/Pi)*elliptic_E(k) in powers of q.

1, -4, 20, -64, 164, -392, 896, -1920, 3908, -7684, 14632, -27072, 48896, -86408, 149760, -255104, 427652, -706568, 1152020, -1855296, 2954056, -4654080, 7260288, -11221632, 17194496, -26131980, 39409960, -59003008, 87728640, -129586568, 190226176, -277587456, 402779396, -581276160, 834539560, -1192216320

Offset: 0

Author

Joerg Arndt, Aug 15 2011

Keywords

• sign

Comments

Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = prod(n>=1, 1-q^(k*n) ) (s=k^2 where k is elliptic k), then the g.f. is hypergeom([-1/2, +1/2], [+1], s) (expansion of 2/Pi*elliptic_E(k) in powers of q).

Examples

			E(k(q)) = 1 - 4*q + 20*q^2 - 64*q^3 + 164*q^4 - 392*q^5 + 896*q^6 - 1920*q^7 +- ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A004018 (elliptic K(k(q))), A115977 (elliptic k(q)^2), A193219.

Programs

Maple

N:= 100: # to get a(0) to a(N)
t3:= curry(JacobiTheta3,0):
t4:= curry(JacobiTheta4,0):
Dq:= f -> q*diff(f,q):
E1:= t3(q)^2:
E2a:= - 2*(t4(q)/t3(q))^2:
E2b:= t4(q)^(-2):
S1:= series(E1,q,N+1):
S2a:= series(E2a,q,N+1):
S2b:= series(Dq(series(E2b,q,N+1)),q,N+1):
S:= series(S1+S2a*S2b,q,N+1):
seq(coeff(S,q,j),j=0..N); # Robert Israel, Sep 02 2015

Mathematica

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ EllipticE[m] / (Pi/2), {q, 0, n}]] (* Michael Somos, Jan 24 2012 *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ -1/2, 1/2, 1, ModularLambda[ Log[q] / (Pi I)]], {q, 0, n}] (* Michael Somos, Jan 24 2012 *)
nmax = 30; dtheta = D[Normal[Series[EllipticTheta[3, 0, x], {x, 0, nmax}]], x]; CoefficientList[Series[(EllipticTheta[4, 0, x]^4 * EllipticTheta[3, 0, x] + 4*x*dtheta) / EllipticTheta[3, 0, x]^3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)
nmax = 30; CoefficientList[4 q D[Log[EllipticTheta[2,0,q]+O[q]^nmax],q]/(EllipticTheta[3,0,q]+O[q]^nmax)^2,q] (* Mamuka Jibladze, Jun 02 2025 *)

Formula

Expansion of theta_3(q)^2 - 2 * (theta_4(q) / theta_3(q))^2 * Dq ( theta_4(q)^-2 ) = theta_3(q)^2 + 4 Dq (theta_4(q)) / (theta_4(q) * theta_3(q)^2) in powers of q where Dq (f) := q * df/dq. - Michael Somos, Jan 24 2012

Expansion of (T4^4 * T3 + 4*q * d/dq T3) / T3^3 where T3 = theta_3(q) and T4 = theta_4(q). - Joerg Arndt, Sep 02 2015

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (Pi * 2^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2019

Expansion of 4 * Dq(theta_2(q))/(theta_2(q) * theta_3(q)^2) in powers of q where Dq(f):= q*df/dq. - Mamuka Jibladze, Jun 02 2025