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A193219 Expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q. 0
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..34.

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 +- ...

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).

Sequence in context: A182039 A174882 A080095 * A213249 A155853 A256552

Adjacent sequences:  A193216 A193217 A193218 * A193220 A193221 A193222

KEYWORD

sign

AUTHOR

Joerg Arndt, Aug 26 2011

STATUS

approved

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Last modified September 24 17:33 EDT 2021. Contains 347651 sequences. (Running on oeis4.)