A193219 Expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q.

1, -2, 8, -16, 18, -32, 112, -192, 0, 62, 1840, -3312, -8320, 16480, 71840, -137280, -522174, 1011392, 4107960, -7945008, -32457600, 62909120, 261338416, -506930112, -2129035776, 4133297534, 17531850576, -34058050240, -145663683072, 283125653280, 1219649036576, -2371704375168, -10281070960128, 20000146662464, 87178011852896

Offset

0,2

Comments

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

The corresponding sequence for sqrt((2/Pi)*elliptic_K(k)) is A000122.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Formula

From Vaclav Kotesovec, Nov 16 2023: (Start)

abs(a(n)) ~ c * d^n / n^(3/2), where

d = 1/sqrt(A072558) = sqrt(A073007) = 3.0477902637682959365706804198489438625220426001497960504423261561153885844...

c = 0.60315114232684465914106139794838284733424313832900503234838172483814652... if n is even and

c = 0.38688142678580145044658710898009855553630625532976316366806686926256857... if n is odd. (End)

Example

sqrt(E(k(q))) = 1 - 2*q + 8*q^2 - 16*q^3 + 18*q^4 - 32*q^5 + 112*q^6 - 192*q^7 +- ...

Mathematica

CoefficientList[Series[Sqrt[(2/Pi) EllipticE[InverseEllipticNomeQ[q]]], {q, 0, 50}], q] (* Jan Mangaldan, Dec 07 2021 *)
nmax = 30; dtheta = D[Normal[Series[EllipticTheta[3, 0, x], {x, 0, nmax}]], x]; CoefficientList[Series[Sqrt[(EllipticTheta[4, 0, x]^4*EllipticTheta[3, 0, x] + 4*x*dtheta)/EllipticTheta[3, 0, x]^3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2023 *)

Crossrefs

Cf. A194094 (elliptic_E(k(q))), A004018 (elliptic_K(k(q))), A000122 (sqrt(elliptic_K(k(q)))=Theta3(q)), A115977 (elliptic k(q)^2).

Cf. A072558, A073007.

Sequence in context: A359228 A174882 A080095 * A213249 A155853 A256552

Adjacent sequences: A193216 A193217 A193218 * A193220 A193221 A193222

Keyword

sign

Author

Joerg Arndt, Aug 26 2011

Status

approved