OFFSET
0,2
COMMENTS
Total number of leaves in all rooted ordered trees with at most n edges. - Michael Somos, Feb 14 2006
Also: Number of UH-free Schroeder paths of semilength n with horizontal steps only at level less than two [see Yan]. - R. J. Mathar, May 24 2008
Hankel transform is A010892. - Paul Barry, Apr 28 2009
Binomial transform of A005773. - Philippe Deléham, Dec 13 2009
Number of vertices all of whose children are leaves in all ordered trees with n+1 edges. Example: a(3) = 15; for an explanation see David Callan's comment in A001519. - Emeric Deutsch, Feb 12 2015
LINKS
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Sherry H. F. Yan, Schröder Paths and Pattern Avoiding Partitions, arXiv:0805.2465 [math.CO], 2008-2009.
FORMULA
a(n) = A079309(n) + 1.
G.f.: 1/((1 - x)*(2 - C)), where C = g.f. for the Catalan numbers A000108. - N. J. A. Sloane, Aug 30 2002
Given g.f. A(x), then x * A(x - x^2) is the g.f. of A024494. - Michael Somos, Feb 14 2006
G.f.: (1 + 1 / sqrt(1 - 4*x)) / (2 - 2*x). - Michael Somos, Feb 14 2006
D-finite with recurrence: n*a(n) - (5*n-2)*a(n-1) + 2*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Dec 02 2012
Remark: The above recurrence is true (it can be easily proved by differentiating the generating function). Notice that it is the same recurrence satisfied by the partial sums of the central binomial coefficients (A006134). - Emanuele Munarini, May 18 2018
0 = a(n)*(16*a(n+1) - 22*a(n+2) + 6*a(n+3)) + a(n+1)*(-18*a(n+1) + 27*a(n+2) - 7*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z if a(n) = 1/2 for n < 0. - Michael Somos, Apr 23 2014
a(n) = ((1 - I/sqrt(3))/2 - binomial(2*n+1, n)*hypergeom([n+3/2, 1], [n+2], 4)). - Peter Luschny, May 18 2018
a(n) = [x^n] 1/((1-x+x^2) * (1-x)^n). - Seiichi Manyama, Apr 06 2024
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 15*x^3 + 50*x^4 + 176*x^5 + 638*x^6 + ...
MAPLE
a := n -> ((1-I/sqrt(3))/2-binomial(2*n+1, n)*hypergeom([n+3/2, 1], [n+2], 4));
seq(simplify(a(n)), n=0..24); # Peter Luschny, May 18 2018
MATHEMATICA
Table[Sum[Binomial[2k-1, k-1], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, May 18 2018 *)
PROG
(PARI) a(n) = (1 + sum(k=0, n, binomial(2*k, k)))/2; \\ Michel Marcus, May 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Name edited by Petros Hadjicostas, Aug 04 2020
STATUS
approved