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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 15 2011
STATUS
approved