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 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

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