A245139 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2).

1, 2, 4, 14, 64, 602, 5344, 58214, 661504, 9666482, 145897984, 2611988414, 47548524544, 1002692887562, 21581168410624, 527328466446614, 13084553110749184, 362312592419199842, 10175324275879051264, 315223836841156264814, 9889646730551557095424, 338833067799589889659322

Offset: 0

Author

Paul D. Hanna, Jul 12 2014

Keywords

• nonn

Comments

Limit (a(n)/n!)^(-1/n) = log(t) = 0.609377863436... where t is the tribonacci constant and satisfies 1 + t + t^2 = t^3.

Examples

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 602*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 4*x^2/2! + 64*x^4/4! + 5344*x^6/6! + 661504*x^8/8! +...
A1(x) = 2*x + 14*x^3/3! + 602*x^5/5! + 58214*x^7/7! + 9666482*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 + 6*x^3/3! + 330*x^5/5! + 21966*x^7/7! + 3507090*x^9/9! + 844747926*x^11/11! + 299180549850*x^13/13! +...
thus A(x)*A(-x) = 1.
		

Crossrefs

Cf. A245138, A245140, A245154, A245165.

Programs

PARI

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(X)) / sqrt(1 - sinh(X)^2*sinh(2*X)^2),n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f.: G(x) * (cosh(x) - sinh(x)*cosh(2*x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2), where G(x) is the e.g.f. of A245140.