A245166 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(2*x)) / (cosh(2*x) - sinh(2*x)*cosh(3*x)).

1, 5, 25, 215, 2425, 33875, 569125, 11160035, 250047025, 6302723075, 176522216125, 5438291613155, 182773714292425, 6654680279353475, 260930805319957525, 10961922511422743075, 491220886240696086625, 23388149451193115459075, 1179066988050425638569325

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 + 5*x + 25*x^2/2! + 215*x^3/3! + 2425*x^4/4! + 33875*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + 2*x + 4*x^2/2! + 62*x^3/3! + 448*x^4/4! + 5882*x^5/5! +...
C(x) = 1 + 3*x + 9*x^2/2! + 63*x^3/3! + 513*x^4/4! + 8043*x^5/5! +...
are the e.g.f.s of A245164 and A245165, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 25*x^2/2! + 2425*x^4/4! + 569125*x^6/6! + 250047025*x^8/8! +...
A1(x) = 5*x + 215*x^3/3! + 33875*x^5/5! + 11160035*x^7/7! + 6302723075*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 5*x + 90*x^3/3! + 8250*x^5/5! + 1946910*x^7/7! + 855018450*x^9/9! + 603621345030*x^11/11! + 624997732481850*x^13/13! +...
thus A(x)*A(-x) = 1.
		

Crossrefs

Cf. A245164, A245165, A245140, A245155.

Cf. A322620, A322190.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[(Cosh[3x]+Sinh[3x]Cosh[2x])/(Cosh[2x]-Sinh[2x]Cosh[3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 21 2024 *)

PARI

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (cosh(2*X) - sinh(2*X)*cosh(3*X)), n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(3*X)) * (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (1 - sinh(2*X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

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

E.g.f.: (cosh(2*x)*cosh(3*x) + sinh(2*x) + sinh(3*x)) / (1 - sinh(2*x)*sinh(3*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 A245164 and A245165, 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(2*x) = (A0(x) + A1(x)*cosh(3*x)) * (cosh(3*x) - sinh(3*x)*A0(x)) / (1 - sinh(3*x)^2*A1(x)^2).

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

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

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

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

FORMULAS FOR TERMS.

a(n) == 0 (mod 5) for n>0.

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

a(n) = Sum_{k=0..n} 2^(n-k) * 3^k * A322620(n,k).

a(n) = Sum_{k=0..n} 2^(n-k) * 3^k * binomial(n,k) * A322190(n,k). (End)