A215364 E.g.f. satisfies: A(x) = 1 + x*A(x)^2*cosh(x*A(x)).

1, 1, 4, 33, 408, 6725, 139200, 3475717, 101722880, 3416079753, 129507425280, 5471712276041, 254965505507328, 12990483544072333, 718474796305989632, 42871067358096134445, 2745230569464318197760, 187780115708775158008337, 13665196427126843296972800

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f. satisfies:

(1) A(x) = (1/x)*Series_Reversion(x-x^2*cosh(x)).

(2) A(x) = 1/(1 - x*A(x)*cosh(x*A(x))).

(3) A(x-x^2*cosh(x)) = 1/(1-x*cosh(x)).

a(n) = [x^n/n!] 1/(1 - x*cosh(x))^(n+1) / (n+1).

a(n) ~ n^(n-1) * s*sqrt(1/(6-2*s-r^2*s^2+r^2*s^3)) / (exp(n) * r^n), where r = 0.2278231894714399793... and s = 1.855593992316816009... are the roots of the equations r*s*(2*cosh(r*s) + r*s*sinh(r*s)) = 1, 1 + r*s^2*cosh(r*s) = s. - Vaclav Kotesovec, Jan 13 2014

Example

E.g.f: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6725*x^5/5! +...

Mathematica

CoefficientList[1/x*InverseSeries[Series[x-x^2*Cosh[x], {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 13 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(1/(1 - x*cosh(x+x*O(x^n)))^(n+1)/(n+1), n)}
(PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A215363, A213644.

Sequence in context: A269926 A331794 A156132 * A370930 A213641 A343673

Adjacent sequences: A215361 A215362 A215363 * A215365 A215366 A215367

Keyword

nonn

Author

Paul D. Hanna, Aug 08 2012

Status

approved