OFFSET
0,5
COMMENTS
Row sums of Riordan array (1,x(1-x^3))^(-1). Also row sums of A124752.
a(n) is the number of ordered trees (A000108) with n vertices in which every non-leaf non-root vertex has exactly two children that are leaves. For example, a(4) counts the 2 trees \ /
| and \|/ . - David Callan, Aug 22 2014
LINKS
J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.
FORMULA
a(n) = ((mod(n,3)+1)*C(4*floor(n/3)+mod(n,3), floor(n/3))/ (3*floor(n/3) + 1 + mod(n, 3))). - Paul Barry, Dec 14 2006
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(w*x)*A(w^2*x), where w = exp(2*Pi*I/3). - Paul D. Hanna, Jun 04 2012
G.f. satisfies: A(x) = 1 + x*A(x)*G(x^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. - Paul D. Hanna, Jun 04 2012
Conjecture: +8019*n*(n-1)*(n+1)*a(n) +17496*n*(n-1)*(n-3)*a(n-1) +2592*(3*n-5)*(n-1)*(3*n-16)*a(n-2) +216*(-224*n^3+48*n^2+3926*n-6331)*a(n-3) +576*(-288*n^3+2448*n^2-6558*n+5443)*a(n-4) +768*(-288*n^3+3600*n^2-14878*n+20375)*a(n-5) -8192*(4*n-23)*(2*n-11)*(4*n-21)*a(n-6)=0. - R. J. Mathar, Oct 30 2014
MAPLE
A124753 := proc(n)
local k, np;
k := modp(n, 3) ;
np := floor(n/3) ;
(k+1)*binomial(np+n, np)/(n+1) ;
end proc:
seq(A124753(n), n=0..40) ; # R. J. Mathar, Oct 30 2014
MATHEMATICA
a[n_] := Module[{q, k}, {q, k} = QuotientRemainder[n, 3]; (k+1)*Binomial[4q + k, q]/(3q + k + 1)];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Nov 20 2017 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*exp(sum(m=1, n\3, 3*polcoeff(log(A+x*O(x^n)), 3*m)*x^(3*m))+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Jun 04 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Nov 06 2006
STATUS
approved