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

0,9

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.

Links

Table of n, a(n) for n=0..68.

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.

Example

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;

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

Crossrefs

Cf. A000108, A114277, A014318.

Sequence in context: A366801 A247498 A091381 * A205106 A351743 A144019

Adjacent sequences: A127153 A127154 A127155 * A127157 A127158 A127159

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 27 2007

Status

approved