Search: id:A010683
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%I A010683
%S A010683 1,2,7,28,121,550,2591,12536,61921,310954,1582791,8147796,
%T A010683 42344121,221866446,1170747519,6216189936,33186295681,178034219986,
%U A010683 959260792775,5188835909516,28167068630713,153395382655222
%N A010683 Let S(x,y) = number of lattice paths from (0,0) to (x,y) that use the 
               step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below 
               y = x. Sequence gives S(n-1,n) = number of `Schroeder' trees with 
               n+1 leaves and root of deg. 2.
%C A010683 a(n) = number of compound propositions "on the negative side" that can 
               be made from n simple propositions.
%C A010683 Convolution of A001003 (the little Schroeder numbers) with itself. - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
%C A010683 Number of dissections of a convex polygon with n+3 sides that have a 
               triangle over a fixed side (the base) of the polygon. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Dec 27 2003
%C A010683 a(n-1) = number of royal paths from (0,0) to (n,n), A006318, with exactly 
               one diagonal step on the line y=x. - David Callan (callan(AT)stat.wisc.edu), 
               Mar 14 2004
%C A010683 Number of short bushes (i.e. ordered trees with no vertices of outdegree 
               1) with n+2 leaves and having root of degree 2. Example: a(2)=7 because, 
               in addition to the five binary trees with 6 edges (they do have 4 
               leaves) we have (i) two edges rb, rc hanging from the root r with 
               three edges hanging from vertex b and (ii) two edges rb, rc hanging 
               from the root r with three edges hanging from vertex c. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Mar 16 2004
%D A010683 Habseiger et al., On the second number of Plutarch, Am. Math. Monthly, 
               105 446 1998.
%D A010683 D. G. Rogers and L. W. Shapiro, "Deques, trees and lattice paths", in 
               Combinatorial Mathematics VIII: Proceedings of the Eighth Australian 
               Conference. Lecture Notes in Mathematics, Vol. 884 (Springer, Berlin, 
               1981), pp. 293-303. Math. Rev., 83g, 05038; Zentralblatt, 469(1982), 
               05005. See Figs. 7a and 8b.
%H A010683 T. D. Noe, <a href="b010683.txt">Table of n, a(n) for n=0..200</a>
%H A010683 E. Pergola and R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Schroeder Triangles, Paths and Parallelogram Polyominoes</
               a>, J. Integer Sequences, 1 (1998), #98.1.7.
%H A010683 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Hipparchus, 
               Plutarch, Schr"oder and Hough</a>, Am. Math. Monthly, Vol. 104, No. 
               4, p. 344, 1997.
%H A010683 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%F A010683 G.f.: ((1-t)^2-(1+t)*sqrt(1-6*t+t^2))/(8*t^2) = (A(t)^2)/x^2, with o.g.f. 
               A(t) of A001003.
%F A010683 a(n)=(2/n)*sum(binomial(n, k)*binomial(n+k+1, k-1), k=1..n) = 2*hypergeom([1-n, 
               n+3], [2], -1), n>=1. a(0)=1. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Sep 12 2005.
%F A010683 G.f.: ((1-t)^2-(1+t)*sqrt(1-6*t+t^2))/(8*t^2) = A(t)^2, with o.g.f. A(t) 
               of A001003.
%t A010683 f[ x_, y_ ] := f[ x, y ] = Module[ {return}, If[ x == 0, return = 1, 
               If[ y == x-1, return = 0, return = f[ x, y-1 ] + Sum[ f[ k, y ], 
               {k, 0, x-1} ] ] ]; return ]; Do[ Print[ Table[ f[ k, j ], {k, 0, 
               j} ] ], {j, 10, 0, -1} ]
%Y A010683 Cf. A001003.
%Y A010683 Right-hand column 2 of triangle A011117.
%Y A010683 Second column of convolution triangle A011117.
%Y A010683 Sequence in context: A150657 A150658 A026770 this_sequence A150659 A150660 
               A150661
%Y A010683 Adjacent sequences: A010680 A010681 A010682 this_sequence A010684 A010685 
               A010686
%K A010683 nonn,nice,easy
%O A010683 0,2
%A A010683 Robert Sulanke (sulanke(AT)diamond.idbsu.edu), N. J. A. Sloane (njas(AT)research.att.com).

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