A245138 E.g.f.: (cosh(x) + sinh(x)*cosh(2*x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2).

1, 1, 1, 13, 49, 361, 3121, 39733, 409249, 6410641, 91979041, 1716516253, 29795642449, 660718214521, 13656276138961, 345794520085573, 8290832204163649, 237409681243284001, 6465138777774530881, 206263448435258395693, 6296129943088315156849, 221484543685548532051081

Offset

0,4

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.

Links

Table of n, a(n) for n=0..21.

Formula

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

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 13*x^3/3! + 49*x^4/4! + 361*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + x^2/2! + 49*x^4/4! + 3121*x^6/6! + 409249*x^8/8! + 91979041*x^10/10! +...
A1(x) = x + 13*x^3/3! + 361*x^5/5! + 39733*x^7/7! + 6410641*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)) = x + 12*x^3/3! + 240*x^5/5! + 24192*x^7/7! + 3452160*x^9/9! + 841961472*x^11/11! + 300389806080*x^13/13! +...
thus A(x)*A(-x) = 1.

Mathematica

With[{nn=30}, CoefficientList[Series[(Cosh[x]+Sinh[x]Cosh[2x])/Sqrt[1-Sinh[x]^2 Sinh[2x]^2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 29 2017 *)

Prog

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

Crossrefs

Cf. A245139, A245140, A245153, A245164.

Sequence in context: A320469 A074014 A147481 * A373620 A197663 A274784

Adjacent sequences: A245135 A245136 A245137 * A245139 A245140 A245141

Keyword

nonn

Author

Paul D. Hanna, Jul 12 2014

Status

approved