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
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (Pi * 2^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2019
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
KEYWORD
sign
AUTHOR
Joerg Arndt, Aug 15 2011
STATUS
approved