1,1

A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1); a double rise in a Grand Dyck path is an occurrence of uu in the path.

Row sums are the central binomial coefficients (A000984). T(n,0)=n+1. T(n,1)=n(n^2-1)/2 (A027480). T(n,2)=(n+1)n(n-1)^2*(n-2)/12 (A027789). Sum(k*T(n,k),k>=0)=(2n-1)!/[n!(n-2)! ] (A000917). For double rises only above the x-axis, see A118964.

T(n,k)=[(n+1)/n]binomial(n,k)binomial(n,k+1). G.f.=G(t,z)=(1+r)^2/(1-tr^2)-1, where r=r(t,z) is the Narayana function, defined by (1+r)(1+tr)z=r, r(t,0)=0. More generally, the g.f. H=H(t,s,u,z), where t,s, and u mark double rises above, below, and on the x-axis, respectively, is H=[1+r(s,z)]/[1-z(1+tr(t,z))(1+ur(s,z))].

Row n is given by seq(binomial(n,i)*binomial(n+2,n+1-i), i=0..n ). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 03 2006

T(3,2)=4 because we have uuuddd, duuudd, dduuud, and ddduuu.

Triangle begins:

2

3, 3

4, 12, 4

5, 30, 30, 5

6, 60, 120, 60, 6

7, 105, 350, 350, 105, 7

8, 168, 840, 1400, 840, 168, 8

9, 252, 1764, 4410, 4410, 1764, 252, 9

r:=(1-z-t*z-sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))/2/t/z: G:=(1+r)^2/(1-t*r^2)-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form

for n from 0 to 10 do seq(binomial(n, i)*binomial(n+2, n+1-i), i=0..n ); od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 03 2006

Cf. A000984, A027480, A027789, A000917, A118964.

Sequence in context: A130743 A111574 A128744 this_sequence A127641 A106821 A065863

Adjacent sequences: A118960 A118961 A118962 this_sequence A118964 A118965 A118966

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), May 07 2006