0,2

a(n) = number of compound propositions "on the negative side" that can be made from n simple propositions.

Convolution of A001003 (the little Schroeder numbers) with itself. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003

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

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

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

Habseiger et al., On the second number of Plutarch, Am. Math. Monthly, 105 446 1998.

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.

T. D. Noe, Table of n, a(n) for n=0..200

E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.

R. P. Stanley, Hipparchus, Plutarch, Schr"oder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.

Index entries for sequences related to trees

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.

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.

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.

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} ]

Cf. A001003.

Right-hand column 2 of triangle A011117.

Second column of convolution triangle A011117.

Sequence in context: A150657 A150658 A026770 this_sequence A150659 A150660 A150661

Adjacent sequences: A010680 A010681 A010682 this_sequence A010684 A010685 A010686

nonn,nice,easy

Robert Sulanke (sulanke(AT)diamond.idbsu.edu), N. J. A. Sloane (njas(AT)research.att.com).