A127156 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D=(1,-1). This definition differs from the one in A091866.

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 5, 2, 3, 3, 1, 19, 7, 5, 6, 4, 1, 67, 26, 12, 11, 10, 5, 1, 232, 93, 38, 23, 21, 15, 6, 1, 804, 325, 131, 61, 44, 36, 21, 7, 1, 2806, 1129, 456, 192, 105, 80, 57, 28, 8, 1, 9878, 3935, 1585, 648, 297, 185, 137, 85, 36, 9, 1, 35072, 13813, 5520

Offset: 0

Author

Emeric Deutsch, Feb 27 2007

Keywords

• nonn
• tabl

Comments

Row sums yield the Catalan numbers (A000108). T(n,0)=A114277(n-3) for n>=3. Sum(k*T(n,k), k=0..n)=A014318(n-1) for n>=1.

Examples

			T(5,2)=5 because we have (UDUD)UUDUDD, (UUDDUUUDDD), (UUUDDDUUDD), (UDUUUUDDDD) and (UUUUDDDDUD) (the initial 2 pyramids are shown between parentheses).
Triangle starts:
1;
0,1;
0,1,1;
1,1,2,1;
5,2,3,3,1;
19,7,5,6,4,1;
		

Crossrefs

Cf. A000108, A114277, A014318.

Programs

Maple

C:=(1-sqrt(1-4*z))/2/z: G:=(1-2*z)*C/(1-z-t*z): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(expand(coeff(Gser,z^n))) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

Formula

G.f.=G(t,z)=(1-2z)C/(1-z-tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,k)=T(n-1,k)+T(n-1,k-1) for n,k>=1.