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A024718 (1/2)*(1 + sum of C(2k,k)) for k = 0,1,2,...,n. 12
1, 2, 5, 15, 50, 176, 638, 2354, 8789, 33099, 125477, 478193, 1830271, 7030571, 27088871, 104647631, 405187826, 1571990936, 6109558586, 23782190486, 92705454896, 361834392116, 1413883873976, 5530599237776, 21654401079326 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..24.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

Sherry H. F. Yan, Schroeder Paths and Pattern Avoiding Partitions, arXiv:0805.2465 [math.CO]

FORMULA

G.f.: 1/((1-x)*(2-C)) where C = g.f. for Catalan numbers A000108. - N. J. A. Sloane, Aug 30 2002

Given g.f. A(x), then x * A(x - x^2) is 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

Conjecture: n*a(n) +(-5*n+2)*a(n-1) +2*(2*n-1)*a(n-2)=0. - R. J. Mathar, Dec 02 2012

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

EXAMPLE

G.f. = 1 + 2*x + 5*x^2 + 15*x^3 + 50*x^4 + 176*x^5 + 638*x^6 + ...

CROSSREFS

Equals A079309(n) + 1. Partial sums of A088218. Bisection of A086905. Second column of triangle A102541.

Sequence in context: A228343 A149949 A149950 * A149951 A157135 A196836

Adjacent sequences:  A024715 A024716 A024717 * A024719 A024720 A024721

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 25 00:39 EST 2018. Contains 299630 sequences. (Running on oeis4.)