A245164 - OEIS

%I #6 Apr 01 2016 17:05:56

%S 1,2,4,62,448,5882,82144,1762742,32401408,839773682,20709251584,

%T 658128799022,19691428538368,735018387765482,26206768383361024,

%U 1124046915311796902,46319665594721763328,2246606049886763789282,105187723831561379774464,5688928855528010885284382

%N E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(3*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2).

%C Limit (a(n)/n!)^(-1/n) = log(t) = 0.3570506972213... where t satisfies 1 + t^2 + t^3 = t^5.

%F E.g.f.: G(x) * (cosh(3*x) - sinh(3*x)*cosh(2*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245166.

%e E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 62*x^3/3! + 448*x^4/4! + 5882*x^5/5! +...

%e Let A(x) = A0(x) + A1(x) where

%e A0(x) = 1 + 4*x^2/2! + 448*x^4/4! + 82144*x^6/6! + 32401408*x^8/8! +...

%e A1(x) = 2*x + 62*x^3/3! + 5882*x^5/5! + 1762742*x^7/7! + 839773682*x^9/9! +...

%e then A0(x)^2 - A1(x)^2 = 1.

%e Note that the logarithm of the e.g.f. is an odd function:

%e Log(A(x)) = 2*x + 54*x^3/3! + 3690*x^5/5! + 1014174*x^7/7! + 421463250*x^9/9! + 303044613894*x^11/11! + 312200620305210*x^13/13! +...

%e thus A(x)*A(-x) = 1.

%t With[{nn=20},CoefficientList[Series[(Cosh[2x]+Sinh[2x]Cosh[3x])/Sqrt[1- Sinh[ 2x]^2 Sinh[3x]^2],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Apr 01 2016 *)

%o (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(3*X)) / sqrt(1 - sinh(2*X)^2*sinh(3*X)^2), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A245165, A245166, A245138, A245153.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 12 2014