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
KEYWORD
sign
AUTHOR
Joerg Arndt, Aug 26 2011
STATUS
approved