A109077 Triangle read by rows: T(n,k) is the number of symmetric Dyck paths of semilength n and having k hills (i.e., peaks at level 1).

1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 0, 1, 0, 1, 6, 1, 2, 0, 0, 1, 13, 0, 5, 0, 1, 0, 1, 22, 2, 6, 2, 2, 0, 0, 1, 46, 0, 16, 0, 6, 0, 1, 0, 1, 80, 6, 24, 4, 6, 3, 2, 0, 0, 1, 166, 0, 58, 0, 19, 0, 7, 0, 1, 0, 1, 296, 18, 90, 13, 26, 6, 6, 4, 2, 0, 0, 1, 610, 0, 211, 0, 71, 0, 22, 0, 8, 0, 1, 0, 1, 1106

Offset: 0

Author

Emeric Deutsch, Jun 17 2005

Keywords

• nonn
• tabl

Comments

Column 0 yields A109078.

T(2n,1)=0, T(2n-1,1) = A000957(n) (the Fine numbers).

Examples

			T(5,2)=2 because we have uduududdud and uduuudddud, where u=(1,1), d=(1,-1).
Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  2, 0, 0, 1;
  4, 0, 1, 0, 1;
  6, 1, 2, 0, 0, 1;
		

Crossrefs

Cf. A000957, A109078.

Programs

Maple

G:=-2*(z+z*sqrt(1-4*z^2)-2*z^2-z*t-1-sqrt(1-4*z^2)+2*z^2*t-z*t*sqrt(1-4*z^2))/(-1-sqrt(1-4*z^2)+2*z)/(-1-sqrt(1-4*z^2)-2*z^2+2*z^2*t^2): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

Formula

G.f.: 2(1 + (t-1)z(1-2z) + q(1 - z + tz))/((1-2z+q)(1+2z^2-2t^2*z^2+q)), where q = sqrt(1 - 4z^2).