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A Catalan-like formula using consecutive odd numbers. Recall that Catalan numbers (A000108) are given by ((n+1)+(n)-1)!/((n+1)!(n)!).

From David Callan, Jun 01 2006: (Start)

a(n) = number of Dyck (2n)-paths (i.e., semilength = 2n) all of whose interior returns to ground level (if any) occur at or before the (2n-2)-nd step, that is, they occur strictly before the midpoint of the path.

For example, a(2)=7 counts UUUUDDDD, UUUDUDDD, UUDUUDDD, UUDUDUDD, UUUDDUDD, UD.UUUDDD, UD.UUDUDD ("." denotes an interior return to ground level).

This result follows immediately from an involution on Dyck paths, due to Emeric Deutsch, defined by E->E, UPDQ -> UQDP (where E is the empty Dyck path; U=upstep, D=downstep and P,Q are arbitrary Dyck paths), because the involution is fixed-point-free on Dyck (2n)-paths and contains one path of the type being counted in each orbit.

a(n) = Sum_{k=0..n-1} C(2n-1-2k)*C(2k). This identity has the following combinatorial interpretation:

a(n) is the number of odd-GL-marked Dyck (2n-1)-paths. An odd-GL vertex is a vertex at location (2i,0) for some odd i >= 1 (path starts at origin). An odd-GL-marked Dyck path is a Dyck path with one of its odd-GL vertices marked. For example, a(2)=7 counts UUUDDD*, UUDUDD*, UD*UUDD, UDUUDD*, UD*UDUD, UDUDUD*, UUDDUD* (the * denotes the marked odd-GL vertex). (End)

a(n+1) = Sum_{k=0..n} C(k)*C(2*n+1-k), n >= 0, with C(n) = A000108(n), also gives the odd part of the bisection of the half-convolution of the Catalan sequence A000108 with itself. For the definition of the half-convolution of a sequence with itself see a comment on A201204. There one also finds the rule for the o.g.f. given below in the formula section. The even part of this bisection is found under A201205. - Wolfdieter Lang, Jan 05 2012

From Peter Bala, Dec 01 2015: (Start)

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = (1/(2*p + q))*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/2) = (1/(4*n + 4))*binomial(4*n + 4, 2*n + 1). Cf. A265101 (Cat(n + 1/3)), A265102 (Cat(n + 1/4)) and A265103 (Cat(n + 1/5)).

Number of maximal faces of the rational associahedron Ass(2*n + 1, 2*n + 3). Number of lattice paths from (0, 0) to (2*n + 3, 2*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (2*n + 1)/(2*n + 3)*x. See Armstrong et al. (End)

Also the number of ordered rooted trees with 2n nodes, most of which are leaves, i.e., the odd bisection of A358585. This follows from Callan's formula below. - Gus Wiseman, Nov 27 2022

There are several versions of a Catalan triangle: see A009766, A008315, A028364.

The subtriangle [1], [2, 3], [5, 7, 9], ..., namely T(N,M-1), for N >= 1, M=1..N, appears as one-point function in the totally asymmetric exclusion process for the parameters alpha=1=beta. See the Derrida et al. and Liggett references given under A067323, where these triangle entries are called T_{N,N+M-1} for the given alpha and beta values. See the row reversed triangle A067323.

Consider a Dyck path as a path with steps N=(0,1) and E=(1,0) from (0,0) to (n,n) that stays weakly above y=x. T(n,m) is the number of Dyck paths of semilength n+1 where the (m+1)st north step is followed by an east step. - Lara Pudwell, Apr 12 2023

In general the half-convolution of a sequence {b(n)}_0^infty with itself is defined by chat(n):=sum(b(k)*b(n-k), k=0..floor(n/2)), n>=0. The o.g.f. of the sequence {chat(n)} is obtained from the bisection 2*chat(2*k) - b(k)^2 = c(2*k), k>=0, with the ordinary convolution c(n):=sum(b(k)*b(n-k),k=0..n), n>=0, and 2*chat(2*k+1) = c(2*k+1), k>=0. This leads to the o.g.f.s for the corresponding even (e) and odd (o) parts:

2*Chate(x) - B2(x) = Ce(x) and 2*Chato(x) = Co(x), where Chate(x):= sum(chat(2*k)*x^k,k=0..infty), Chato(x):= sum(chat(2*k+1)*x^k,k=0..infty), B2(x) := sum(b(k)^2*x^k, k=0..infty), Ce(x) := sum(c(2*k)*x^k, k=0..infty) and Co(x) := sum(c(2*k+1)*x^k, k=0..infty). Thus Chate(x)=(Ce(x) + B2(x))/2 and Chato(x)=Co(x)/2. Expressing this in terms of C(x), the o.g.f. of {c(n)}, and B2(x) leads to the result: Chat(x)= (C(x) + B2(x^2))/2.

In the Catalan case b(n)=A000108(n), c(n)=b(n+1), C(x)= (cata(x)+1)/x, with the o.g.f. of A000108 cata(x)=(1-sqrt(1-4*x))/(2*x), and B2(x) is found under A001246 to be (-1 + hypergeom([-1/2,-1/2],[1],16*x))/(4*x). This produces the o.g.f. given in the formula section.

This computation was motivated by a question about the o.g.f. of A000992 ("half-Catalan numbers"). Note, however, that this sequence is not the half-convolution of the Catalan numbers presented here.

Apparently the number of hills to the left of or at the midpoint in all Dyck paths of semilength n+1. [David Scambler, Apr 30 2013]

a(N,p) equals X_{N}(N+1,p) := T_{N,p} for alpha= 1 =beta and N>=p>=1 in the Derrida et al. 1992 reference. The one-point correlation functions A000108(n)%20(Catalan)%20in%20this%20reference.%20See%20also%20the%20Derrida%20et%20al.%201993%20reference.%20In%20the%20Liggett%201999%20reference%20mu">{N} for alpha= 1 =beta equal a(N,K)/C(N+1) with C(n)=A000108(n) (Catalan) in this reference. See also the Derrida et al. 1993 reference. In the Liggett 1999 reference mu{N}{eta:eta(k)=1} of prop. 3.38, p. 275 is identical with _{N} and rho=0 and lambda=1.

Identity for each row n>=1: a(n,m)+a(n,n-m+1)= C(n+1), with C(n+1)=A000108(n+1)(Catalan) for every m=1..floor((n+1)/2). E.g., a(2k+1,k+1)=C(2*(k+1)).

The first column sequences (diagonals of A028364) are: A000108(n+1), A000245, A067324-6 for m=0..4.