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

0,7

Comments

Column 0 yields A109078.

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

Links

Table of n, a(n) for n=0..91.

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).

Example

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;

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

Crossrefs

Cf. A000957, A109078.

Sequence in context: A091866 A168511 A111146 * A378178 A355916 A309023

Adjacent sequences: A109074 A109075 A109076 * A109078 A109079 A109080

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 17 2005

Status

approved