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

%I A193219 #35 Feb 15 2025 20:26:56
%S A193219 1,-2,8,-16,18,-32,112,-192,0,62,1840,-3312,-8320,16480,71840,-137280,
%T A193219 -522174,1011392,4107960,-7945008,-32457600,62909120,261338416,
%U A193219 -506930112,-2129035776,4133297534,17531850576,-34058050240,-145663683072,283125653280,1219649036576,-2371704375168,-10281070960128,20000146662464,87178011852896
%N A193219 Expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q.
%C A193219 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).
%C A193219 The corresponding sequence for sqrt((2/Pi)*elliptic_K(k)) is A000122.
%H A193219 Vaclav Kotesovec, <a href="/A193219/b193219.txt">Table of n, a(n) for n = 0..2000</a>
%F A193219 From _Vaclav Kotesovec_, Nov 16 2023: (Start)
%F A193219 abs(a(n)) ~ c * d^n / n^(3/2), where
%F A193219 d = 1/sqrt(A072558) = sqrt(A073007) = 3.0477902637682959365706804198489438625220426001497960504423261561153885844...
%F A193219 c = 0.60315114232684465914106139794838284733424313832900503234838172483814652... if n is even and
%F A193219 c = 0.38688142678580145044658710898009855553630625532976316366806686926256857... if n is odd. (End)
%e A193219 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 +- ...
%t A193219 CoefficientList[Series[Sqrt[(2/Pi) EllipticE[InverseEllipticNomeQ[q]]], {q, 0, 50}], q] (* _Jan Mangaldan_, Dec 07 2021 *)
%t A193219 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 *)
%Y A193219 Cf. A194094 (elliptic_E(k(q))), A004018 (elliptic_K(k(q))), A000122 (sqrt(elliptic_K(k(q)))=Theta3(q)), A115977 (elliptic k(q)^2).
%Y A193219 Cf. A072558, A073007.
%K A193219 sign
%O A193219 0,2
%A A193219 _Joerg Arndt_, Aug 26 2011