A121529 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an odd level (n >= 1, k >= 0).

1, 1, 1, 1, 4, 1, 10, 2, 1, 19, 14, 1, 33, 50, 5, 1, 55, 132, 45, 1, 90, 301, 205, 13, 1, 146, 631, 680, 139, 1, 236, 1255, 1892, 763, 34, 1, 381, 2409, 4717, 3019, 419, 1, 615, 4509, 10920, 9846, 2677, 89, 1, 993, 8283, 23974, 28292, 12241, 1241, 1, 1604, 14998

Offset: 1

Author

Emeric Deutsch, Aug 05 2006

Keywords

• nonn
• tabf

Comments

A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

Row n contains 1+floor(n/2) terms.

Row sums are the odd-indexed Fibonacci numbers (A001519).

T(2n,n) = Fibonacci(2n-1) (A001519).

Sum_{k>=0} k*T(n,k) = A121530(n).

Examples

			T(4,2)=2 because we have U/UDDU/UDD and U/UU/UDDDD, where U=(1,1) and D=(1,-1) (the double rises at an odd level are indicated by a /).
Triangle starts:
  1;
  1,  1;
  1,  4;
  1, 10,  2;
  1, 19, 14;
  1, 33, 50,  5;
		

Links

• E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

Crossrefs

Cf. A001519, A121530, A121531, A054142.

Programs

Maple

G:=z*(1-t*z^2)*(1-z+t*z-z^2-t*z^2-t^2*z^3)/(1-z-t*z^2)/(1-z-z^2-3*t*z^2-t*z^3+t^2*z^4): Gser:=simplify(series(G,z=0,18)): for n from 1 to 15 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 15 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

Formula

G.f.: G(t,z) = z*(1-tz^2)*(1 - z + tz - z^2 - tz^2 - t^2*z^3)/((1 - z - tz^2)*(1 - z - z^2 - 3tz^2 - tz^3 + t^2*z^4)).