login
This site is supported by donations to The OEIS Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A194094 Expansion of (2/Pi)*elliptic_E(k) in powers of q. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

LINKS

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

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

EXAMPLE

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

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 *)

CROSSREFS

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

Sequence in context: A226424 A225260 A131479 * A055538 A302317 A319779

Adjacent sequences:  A194091 A194092 A194093 * A194095 A194096 A194097

KEYWORD

sign

AUTHOR

Joerg Arndt, Aug 15 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)