A245165 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(2*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2).

1, 3, 9, 63, 513, 8043, 115209, 2170983, 42235713, 1075192083, 27302385609, 837303386703, 25799446123713, 938330441750523, 34249273199668809, 1436790115786367223, 60444494320614768513, 2873965406506938435363, 137038195324637653852809, 7283819678458854655944543

Offset

0,2

Comments

Limit (a(n)/n!)^(-1/n) = log(t) = 0.3570506972213... where t satisfies 1 + t^2 + t^3 = t^5.

Links

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

Formula

E.g.f.: G(x) * (cosh(2*x) - sinh(2*x)*cosh(3*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245166.

Example

E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 63*x^3/3! + 513*x^4/4! + 8043*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 9*x^2/2! + 513*x^4/4! + 115209*x^6/6! + 42235713*x^8/8! +...
A1(x) = 3*x + 63*x^3/3! + 8043*x^5/5! + 2170983*x^7/7! + 1075192083*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)) = 3*x + 36*x^3/3! + 4560*x^5/5! + 932736*x^7/7! + 433555200*x^9/9! + 300576731136*x^11/11! +...
thus A(x)*A(-x) = 1.

Mathematica

With[{nn=20}, CoefficientList[Series[(Cosh[3x]+Sinh[3x]Cosh[2x])/Sqrt[1-Sinh[ 2x]^2 Sinh[3x]^2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Apr 27 2021 *)

Prog

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

Crossrefs

Cf. A245164, A245166, A245139, A245154.

Sequence in context: A121696 A108557 A109285 * A091760 A144525 A276535

Adjacent sequences: A245162 A245163 A245164 * A245166 A245167 A245168

Keyword

nonn

Author

Paul D. Hanna, Jul 12 2014

Status

approved