A194958 E.g.f. satisfies: A(x) = cosh(x) + x*A(x)^2.

1, 1, 5, 36, 409, 6280, 121501, 2839424, 77834737, 2449360512, 87040440181, 3447798906112, 150645874207753, 7197909122453504, 373365727806824845, 20895734364795187200, 1255062315134651501281, 80528111291313595580416, 5497183726333878664852453

Offset

0,3

Comments

The radius of convergence r of the e.g.f. A(x) satisfies: r = 1/(4*cosh(r)) = limit (n+1)*a(n)/a(n+1) = 0.24280736240... with A(r) = 1/(2*r) = 2.059245630...

Links

G. C. Greubel, Table of n, a(n) for n = 0..360

Formula

E.g.f.: A(x) = (1 - sqrt(1 - 4*x*cosh(x))) / (2*x).

a(n) = (1 + (-1)^n)/2 + n*Sum_{k=0..n-1} C(n-1,k)*a(k)*a(n-1-k) for n>=0.

a(n) ~ n! * sqrt(cosh(r)+r*sinh(r))/(2*r^(n+1/2)*sqrt(Pi)*n^(3/2)), where r = 0.2428073624... is defined in the comment. - Vaclav Kotesovec, Sep 21 2013

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 36*x^3/3! + 409*x^4/4! + 6280*x^5/5! +...
Related expansion:
A(x)^2 = 1 + 2*x + 12*x^2/2! + 102*x^3/3! + 1256*x^4/4! + 20250*x^5/5! +...

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*x*Cosh[x]])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 21 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*cosh(x +O(x^(n+2))))) / (2*x), n)}
(PARI) {a(n)=(1 + (-1)^n)/2 +n*sum(k=0, n-1, binomial(n-1, k)*a(k)*a(n-1-k))}

Crossrefs

Cf. A194957, A194471.

Sequence in context: A356001 A230887 A365356 * A241346 A132686 A322180

Adjacent sequences: A194955 A194956 A194957 * A194959 A194960 A194961

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2011

Status

approved