OFFSET
0,3
COMMENTS
The related sequence A007889 enumerates the number of intransitive (or alternating) trees.
a(n+1) is the number of incomplete ternary trees on n labeled vertices in which each left child has a larger label than its parent and each middle child has a smaller label than its parent. - Brian Drake, Jul 28 2008
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..363
FORMULA
a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+k+1)^(n-1) - Vladeta Jovovic, Mar 31 2008.
E.g.f. satisfies: A( 2*x/( exp(x) + exp(2*x) ) ) = exp(x).
E.g.f.: A(x) = inverse function of 2*log(x)/(x + x^2).
E.g.f.: A(x) = exp( Series_Reversion[ 2*x/(exp(x) + exp(2*x)) ] ).
E.g.f.: A(x) = G(x/2) where G(x) = e.g.f. of A138764.
More generally, if A(x) = Sum_{n>=0} a(n)*x^n/n! = exp( x*[A(x) + A(x)^m]/2 ) then a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+(m-1)*k+1)^(n-1) and if B(x) = Sum_{n>=0} b(n)*x^n/n! = log(A(x)) then b(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+(m-1)*k)^(n-1). - Paul D. Hanna and Vladeta Jovovic, Apr 02 2008
Powers of e.g.f.: If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then
. a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(n+k+p)^(n-1).
Given e.g.f. A(x), let B(x) = e.g.f. of A007889, then
. A(x) = B(x*A(x)) = (1/x)*Series_Reversion(x/B(x)) and
. B(x) = A(x/B(x)) = x/Series_Reversion(x*A(x)).
a(n) ~ n^(n-1)*(1+r)^n*r^(n+1)/(sqrt(1+3*r)*(1-r)^(2*n+1)*exp(n)*2^n), where r = 0.6472709258412625... is the root of the equation (r/(1-r))^(1+r) = e. - Vaclav Kotesovec, Jun 15 2013
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 364*x^4/4! + 5766*x^5/5! + ...
MATHEMATICA
Table[1/2^n * Sum[Binomial[n, k]*(n+k+1)^(n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 15 2013 *)
PROG
(PARI) a(n)=(1/2^n)*sum(k=0, n, binomial(n, k)*(n+k+1)^(n-1))
(PARI) /* Series Reversion: */
a(n)=local(X=x+x*O(x^n)); n!*polcoeff(exp(serreverse(2*x/(exp(X)+exp(2*X)) )), n)
(PARI) /* Coefficients of A(x)^p are given by: */
{a(n, p=1)=(1/2^n)*sum(k=0, n, binomial(n, k)*p*(n+k+p)^(n-1))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 01 2008, Apr 02 2008, Apr 03 2008
STATUS
approved