A245140 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(2*x)).

1, 3, 9, 45, 297, 2433, 23949, 275145, 3612177, 53348193, 875453589, 15802999545, 311196040857, 6638817262353, 152521855979229, 3754366520240745, 98575724288354337, 2749997026637342913, 81230299711952152869, 2532707187355266614745, 83124358113443446120617, 2864579803637260793877873

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.

In general, the radius of convergence r of the e.g.f. (cosh(p*x) + sinh(p*x)*cosh(q*x)) / (cosh(q*x) - sinh(q*x)*cosh(p*x)), where p and q are positive integers, equals r = log(t) such that t is the positive real root that satisfies: 1 + t^p + t^q = t^(p+q).

Examples

			E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 45*x^3/3! + 297*x^4/4! + 2433*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + x + x^2/2! + 13*x^3/3! + 49*x^4/4! + 361*x^5/5! + 3121*x^6/6! +...
C(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 602*x^5/5! + 5344*x^6/6! +...
are the e.g.f.s of A245138 and A245139, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 9*x^2/2! + 297*x^4/4! + 23949*x^6/6! + 3612177*x^8/8! +...
A1(x) = 3*x + 45*x^3/3! + 2433*x^5/5! + 275145*x^7/7! + 53348193*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 3*x + 18*x^3/3! + 570*x^5/5! + 46158*x^7/7! + 6959250*x^9/9! + 1686709398*x^11/11! + 599570355930*x^13/13! + 3754366520240745*x^15/15! +...
thus A(x)*A(-x) = 1.
		

Crossrefs

Cf. A245138, A245139, A245155, A245166.

Cf. A322620, A322190.

Programs

PARI

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

PARI

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

Formula

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

E.g.f.: (cosh(x)*cosh(2*x) + sinh(x) + sinh(2*x)) / (1 - sinh(x)*sinh(2*x)). - Paul D. Hanna, Dec 22 2018

E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245138 and A245139, respectively.

Let e.g.f. A(x) = A0(x) + A1(x) where A0(x) = (A(x)+A(-x))/2 and A1(x) = (A(x)-A(-x))/2, then:

(1) A0(x)^2 - A1(x)^2 = 1.

(2) exp(x) = (A0(x) + A1(x)*cosh(2*x)) * (cosh(2*x) - sinh(2*x)*A0(x)) / (1 - sinh(2*x)^2*A1(x)^2).

(3) exp(2*x) = (A0(x) + A1(x)*cosh(x)) * (cosh(x) - sinh(x)*A0(x)) / (1 - sinh(x)^2*A1(x)^2).

From Paul D. Hanna, Dec 22 2018: (Start)

(4) exp(x) = (A0(x)*cosh(2*x) + A1(x) - sinh(2*x)) / (1 + sinh(2*x)*A1(x)).

(5) exp(2*x) = (A0(x)*cosh(x) + A1(x) - sinh(x)) / (1 + sinh(x)*A1(x)). (End)

FORMULAS FOR TERMS.

From Paul D. Hanna, Dec 22 2018: (Start)

a(n) = Sum_{k=0..n} 2^k * A322620(n,k).

a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * A322190(n,k). (End)