OFFSET
0,2
REFERENCES
E. van Fossen Conrad, Some continued fraction expansions of elliptic functions, PhD thesis, The Ohio State University, 2002, p. 35. [Paul Barry, Mar 29 2010]
FORMULA
G.f. satisfies: A(x)*A(i*x) = 1 where A(x) = Sum_{n>=0} a(n)*x^(2n)/(2n)! and i^2=-1.
From Paul D. Hanna, Apr 30 2009: (Start)
The o.g.f. G(x), as the formal Laplace transform of e.g.f. cm4(x)^2 + sm4(x)^2, is given by the continued fraction:
G(x) = 1/(1-2(x)^2/(1-(2x)^2/(1-2(3x)^2/(1-(4x)^2/(1-2(5x)^2/(1-...)))))).
(End)
Let f(x) = sqrt(x^4-1). Let D be the operator f(x)*d/dx. Then it appears that D^(2*n-1)(f(x)) evaluated at x = 1 equals a(n) (checked up to a(14)). - Peter Bala, Aug 30 2011
G.f.: 1/Q(0), where Q(k)= 1 - 2*x*(2*k+1)^2/(1 - x*(2*k+2)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
E.g.f.: A(x) = d/dx Series_Reversion( Integral sqrt( cosh(2*x) ) dx ). - Paul D. Hanna, Aug 13 2017
EXAMPLE
E.g.f.: A(x) = 1 + 2*x^2/2! + 12*x^4/4! + 216*x^6/6! + 7056*x^8/8! + 368928*x^10/10! + ...
From Paul D. Hanna, Apr 30 2009: (Start)
O.g.f.: G(x) = 1 + 2*x^2 + 12*x^4 + 216*x^6 + 7056*x^8 + ...
G(x) = 1/(1 - 2x^2/(1 - 4x^2/(1 - 18x^2/(1 - 16x^2/(1 - 50x^2/(1-...)))))).
(End)
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], With[ {m = 2 n}, m! SeriesCoefficient[ JacobiND[ x, 2], {x, 0, m}]]]; (* Michael Somos, Oct 18 2011 *)
a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, (-1)^n m! SeriesCoefficient[ JacobiAmplitude[ x, 2], {x, 0, m}]]]; (* Michael Somos, Mar 13 2017 *)
Table[Abs[SeriesCoefficient[InverseSeries[Series[EllipticF[x, 2], {x, 0, 40}]], 2 n + 1] (2 n + 1)!], {n, 0, 19}] (* Benedict W. J. Irwin, Apr 04 2017 *)
nmax = 20; s = CoefficientList[Series[JacobiNC[Sqrt[2] x, 1/2], {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!; Table[s[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Nov 29 2020 *)
PROG
(PARI) {a(n) = my(A); if(n<0, 0, A=x*O(x); for(i=0, n, A = 1 + intformal( intformal(A^3)^3 ) ); (2*n)!*polcoeff( A^2 + sqrt(A^4-1), 2*n))}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = deriv( serreverse( intformal( 1/sqrt(cosh(2*x + O(x^(2*n+2)))) ))); (2*n)!*polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 13 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 02 2009
STATUS
approved