A153302 - OEIS

%I #39 Dec 01 2020 20:45:50

%S 1,2,12,216,7056,368928,28340928,3000945024,419025809664,

%T 74600006164992,16492933524114432,4433180509950990336,

%U 1423737921326106710016,538417241668323364773888,236818870322157143631249408

%N G.f.: A(x) = cm4(x)^2 + sm4(x)^2 where cm4(x) and sm4(x) are the g.f.s of A153300 and A153301, respectively, that satisfy cm4(x)^4 - sm4(x)^4 = 1.

%D 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]

%F G.f. satisfies: A(x)*A(i*x) = 1 where A(x) = Sum_{n>=0} a(n)*x^(2n)/(2n)! and i^2=-1.

%F From _Paul D. Hanna_, Apr 30 2009: (Start)

%F 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:

%F G(x) = 1/(1-2(x)^2/(1-(2x)^2/(1-2(3x)^2/(1-(4x)^2/(1-2(5x)^2/(1-...)))))).

%F (End)

%F 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

%F 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

%F E.g.f.: A(x) = d/dx Series_Reversion( Integral sqrt( cosh(2*x) ) dx ). - _Paul D. Hanna_, Aug 13 2017

%e 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! + ...

%e From _Paul D. Hanna_, Apr 30 2009: (Start)

%e O.g.f.: G(x) = 1 + 2*x^2 + 12*x^4 + 216*x^6 + 7056*x^8 + ...

%e G(x) = 1/(1 - 2x^2/(1 - 4x^2/(1 - 18x^2/(1 - 16x^2/(1 - 50x^2/(1-...)))))).

%e (End)

%t 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 *)

%t 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 *)

%t 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 *)

%t 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 *)

%o (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))}

%o for(n=0, 20, print1(a(n), ", "))

%o (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)}

%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 13 2017

%Y Cf. A153300, A153301.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 02 2009