A200318 E.g.f. satisfies: A(x) = x-1 + cosh(A(x)).

1, 1, 3, 16, 120, 1156, 13608, 189316, 3039060, 55291336, 1124309208, 25268818576, 622008616320, 16642670404816, 480923246983728, 14926731083999296, 495243684302520000, 17491488288340789696, 655224017429959987968, 25947019896579324410176, 1083050878686674070676800

Offset

1,3

Comments

a(n) is the number of leaf labeled rooted trees with n leaves in which the outdegrees of the root and all internal nodes are positive even integers. - Geoffrey Critzer, Jul 31 2016

Links

Table of n, a(n) for n=1..21.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 128,(labeled hierarchies).

Formula

E.g.f. satisfies:

(1) A(x) = Series_Reversion(1+x - cosh(x)).

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n / n!.

(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n/x / n! ).

a(n) ~ n^(n-1) / (2^(1/4) * exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014

Example

E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 120*x^5/5! +...
where A(1+x - cosh(x)) = x and A(x) = x-1 + cosh(A(x)).
The e.g.f. satisfies:
A(x) = x + (cosh(x)-1) + d/dx (cosh(x)-1)^2/2! + d^2/dx^2 (cosh(x)-1)^3/3! + d^3/dx^3 (cosh(x)-1)^4/4! +...
as well as the logarithmic series:
log(A(x)/x) = (cosh(x)-1)/x + d/dx (cosh(x)-1)^2/x/2! - d^2/dx^2 (cosh(x)-1)^3/x/3! + d^3/dx^3 (cosh(x)-1)^4/x/4! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[1 + x - Cosh[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(1+x-cosh(x+x^2*O(x^n))), n)}
for(n=1, 21, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Crossrefs

Cf. A200317, A000311, A052526.

Sequence in context: A192783 A136168 A187735 * A120015 A003692 A166883

Adjacent sequences: A200315 A200316 A200317 * A200319 A200320 A200321

Keyword

nonn

Author

Paul D. Hanna, Nov 15 2011

Status

approved