This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A194094 #41 Jun 19 2025 00:06:57
%S A194094 1,-4,20,-64,164,-392,896,-1920,3908,-7684,14632,-27072,48896,-86408,
%T A194094 149760,-255104,427652,-706568,1152020,-1855296,2954056,-4654080,
%U A194094 7260288,-11221632,17194496,-26131980,39409960,-59003008,87728640,-129586568,190226176,-277587456,402779396,-581276160,834539560,-1192216320
%N A194094 Expansion of (2/Pi)*elliptic_E(k) in powers of q.
%C A194094 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).
%H A194094 Robert Israel, <a href="/A194094/b194094.txt">Table of n, a(n) for n = 0..2000</a>
%F A194094 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
%F A194094 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
%F A194094 a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (Pi * 2^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2019
%F A194094 Expansion of 4 * Dq(theta_2(q))/(theta_2(q) * theta_3(q)^2) in powers of q where Dq(f):= q*df/dq. - _Mamuka Jibladze_, Jun 02 2025
%e A194094 E(k(q)) = 1 - 4*q + 20*q^2 - 64*q^3 + 164*q^4 - 392*q^5 + 896*q^6 - 1920*q^7 +- ...
%p A194094 N:= 100: # to get a(0) to a(N)
%p A194094 t3:= curry(JacobiTheta3,0):
%p A194094 t4:= curry(JacobiTheta4,0):
%p A194094 Dq:= f -> q*diff(f,q):
%p A194094 E1:= t3(q)^2:
%p A194094 E2a:= - 2*(t4(q)/t3(q))^2:
%p A194094 E2b:= t4(q)^(-2):
%p A194094 S1:= series(E1,q,N+1):
%p A194094 S2a:= series(E2a,q,N+1):
%p A194094 S2b:= series(Dq(series(E2b,q,N+1)),q,N+1):
%p A194094 S:= series(S1+S2a*S2b,q,N+1):
%p A194094 seq(coeff(S,q,j),j=0..N); # _Robert Israel_, Sep 02 2015
%t A194094 a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ EllipticE[m] / (Pi/2), {q, 0, n}]] (* _Michael Somos_, Jan 24 2012 *)
%t A194094 a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ -1/2, 1/2, 1, ModularLambda[ Log[q] / (Pi I)]], {q, 0, n}] (* _Michael Somos_, Jan 24 2012 *)
%t A194094 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 *)
%t A194094 nmax = 30; CoefficientList[4 q D[Log[EllipticTheta[2,0,q]+O[q]^nmax],q]/(EllipticTheta[3,0,q]+O[q]^nmax)^2,q] (* _Mamuka Jibladze_, Jun 02 2025 *)
%Y A194094 Cf. A004018 (elliptic K(k(q))), A115977 (elliptic k(q)^2), A193219.
%K A194094 sign
%O A194094 0,2
%A A194094 _Joerg Arndt_, Aug 15 2011