A245164 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(3*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2).

1, 2, 4, 62, 448, 5882, 82144, 1762742, 32401408, 839773682, 20709251584, 658128799022, 19691428538368, 735018387765482, 26206768383361024, 1124046915311796902, 46319665594721763328, 2246606049886763789282, 105187723831561379774464, 5688928855528010885284382

Offset: 0

Author

Paul D. Hanna, Jul 12 2014

Keywords

• nonn

Comments

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

Examples

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 62*x^3/3! + 448*x^4/4! + 5882*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 4*x^2/2! + 448*x^4/4! + 82144*x^6/6! + 32401408*x^8/8! +...
A1(x) = 2*x + 62*x^3/3! + 5882*x^5/5! + 1762742*x^7/7! + 839773682*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm of the e.g.f. is an odd function:
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! +...
thus A(x)*A(-x) = 1.
		

Crossrefs

Cf. A245165, A245166, A245138, A245153.

Programs

Mathematica

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 *)

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)}
for(n=0, 30, print1(a(n), ", "))

Formula

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.