A121481 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at odd level (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 3, 3, 6, 0, 1, 5, 14, 5, 9, 0, 1, 12, 22, 35, 7, 12, 0, 1, 22, 68, 53, 65, 9, 15, 0, 1, 49, 127, 203, 97, 104, 11, 18, 0, 1, 94, 329, 390, 444, 153, 152, 13, 21, 0, 1, 201, 664, 1157, 873, 816, 221, 209, 15, 24, 0, 1, 396, 1576, 2456, 2925, 1627, 1345

Offset

0,8

Comments

Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,0)=A121482(n). Sum(k*T(n,k),k=0..n)=A121483(n).

Links

Table of n, a(n) for n=0..71.

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

Formula

G.f.: G(t,z) = (tz^2+z^2+z-1)(tz^3+2z^2-1)/(1-4z^2-z-tz+2tz^4+4z^4-z^6+2z^3+tz^3+t^2*z^3).

Example

T(3,1)=3 because we have U|DUUDUDD, UUDUU|DDD and UUU|DDUDD, where U=(1,1) and D=(1,-1) (the peaks at odd level are shown by a |; the Dyck path UUDUDDUD has 1 peak at odd level but it is not nondecreasing).
Triangle starts:
1;
0,1;
1,0,1;
1,3,0,1;
3,3,6,0,1;
5,14,5,9,0,1;

Maple

G:=(t*z^2+z^2+z-1)*(t*z^3+2*z^2-1)/(1-4*z^2-z-t*z+2*z^4*t+4*z^4-z^6+2*z^3+t*z^3+z^3*t^2): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(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. A001519, A121482, A121483, A121484.

Sequence in context: A207032 A169940 A279010 * A366875 A374361 A121469

Adjacent sequences: A121478 A121479 A121480 * A121482 A121483 A121484

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 02 2006

Status

approved