OFFSET
0,3
LINKS
Y. Sun and Z. Wang, Consecutive pattern avoidances in non-crossing trees, Graph. Combinat. 26 (2010) 815-832.
FORMULA
G.f.: (3-sqrt(5-4*C(x)))/2, where C(x) is the generating function of the Catalan numbers.
a(n) = Sum_(k=1..n, (C(2*k-2,k-1)*Sum_(i=k..n, i*C(i-1,k-1) * C(2*n-i-1,n-1))) /k)/n, n>0, a(0)=1. - Vladimir Kruchinin, Jan 23 2013
D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(-27*n^2+47*n-16)*a(n-1) +30*(17*n^2-63*n+56)*a(n-2) -500*(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Jan 24 2013
a(n) = Sum_{k=1..n} binomial(2*k-2,k-1)*binomial(2*n,n-k)/n, a(0)=1. - Vladimir Kruchinin, Apr 28 2016
a(n) = Catalan(n)*hypergeom([1/2, 1-n], [n+2], -4) for n>=1. - Peter Luschny, Apr 28 2016
G.f.: A=A(x) satisfies 0 = -x*A^4 + 6*x*A^3 + (-11*x - 1)*A^2 + (6*x + 3)*A + (-x - 2). - Joerg Arndt, Apr 29 2016
G.f.: A(x) = F(G(x)), where F(x) = 1 + x*C(x), G(x) = x*C(x)^2, and C(x) is the Catalan generating function. - Alexander Burstein, Nov 10 2021
MAPLE
a := n -> `if`(n=0, 1, hypergeom([1/2, 1-n], [n+2], -4)*binomial(2*n, n)/(n+1)):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Apr 28 2016
MATHEMATICA
a[n_] := Sum[Binomial[2k-2, k-1] Binomial[2n, n-k]/n, {k, 1, n}]; a[0] = 1;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 23 2019, after Vladimir Kruchinin *)
PROG
(PARI)
N = 66; x = 'x + O('x^N);
C=(1-sqrt(1-4*x))/(2*x);
gf = (3-sqrt(5-4*C))/2;
v = Vec(gf)
/* Joerg Arndt, Jan 04 2013 */
(Maxima)
a(n):=if n=0 then 1 else sum((binomial(2*k-2, k-1) * sum(i*binomial(i-1, k-1) * binomial(2*n-i-1, n-1), i, k, n)) / k, k, 1, n) / n; /* Vladimir Kruchinin, Jan 23 2013 */
a(n):=if n=0 then 1 else sum(binomial(2*k-2, k-1)*binomial(2*n, n-k), k, 1, n)/n; /* Vladimir Kruchinin, Apr 28 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jan 04 2013
STATUS
approved