A245155 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(3*x)).

1, 4, 16, 100, 832, 8644, 107776, 1567780, 26063872, 487466884, 10129985536, 231560895460, 5774444019712, 155997355725124, 4538464905527296, 141469868440031140, 4703786933664612352, 166172927821116399364, 6215792183431115309056, 245422172388559255422820

Offset

0,2

Comments

Limit (a(n)/n!)^(-1/n) = log(t) = 0.4812118250596... where t = (1+sqrt(5))/2 satisfies 1 + t + t^3 = t^4.

Links

Table of n, a(n) for n=0..19.

Formula

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

E.g.f.: (cosh(x)*cosh(3*x) + sinh(x) + sinh(3*x)) / (1 - sinh(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 A245153 and A245154, 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(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(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(3*x) + A1(x) - sinh(3*x)) / ((1 + sinh(3*x)*A1(x)).

(5) exp(3*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} 3^k * A322620(n,k).

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

a(n) ~ 2*sqrt(2*Pi/5) * n^(n+1/2) / (exp(n) * (log((1+sqrt(5))/2))^(n+1)). - Vaclav Kotesovec, Nov 04 2014

Example

E.g.f.: A(x) = 1 + 4*x + 16*x^2/2! + 100*x^3/3! + 832*x^4/4! + 8644*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5!  + 12421*x^6/6! +...
C(x) = 1 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! + 26109*x^6/6! +...
are the e.g.f.s of A245153 and A245154, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 16*x^2/2! + 832*x^4/4! + 107776*x^6/6! + 26063872*x^8/8! +...
A1(x) = 4*x + 100*x^3/3! + 8644*x^5/5! + 1567780*x^7/7! + 487466884*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 4*x + 36*x^3/3! + 1860*x^5/5! + 240996*x^7/7! + 58280580*x^9/9! + 22651336356*x^11/11! + 12912049359300*x^13/13! + 10148316042271716*x^15/15! +...
thus A(x)*A(-x) = 1.

Prog

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

Crossrefs

Cf. A245153, A245154, A245140, A245166.

Cf. A322620, A322190.

Sequence in context: A074187 A111883 A370529 * A334750 A094637 A332783

Adjacent sequences: A245152 A245153 A245154 * A245156 A245157 A245158

Keyword

nonn

Author

Paul D. Hanna, Jul 12 2014

Status

approved