login
A121531
Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an even level (n >= 1, k >= 0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
2
1, 2, 4, 1, 7, 6, 12, 20, 2, 20, 51, 18, 33, 115, 80, 5, 54, 240, 262, 54, 88, 477, 725, 294, 13, 143, 916, 1803, 1158, 161, 232, 1716, 4170, 3768, 1026, 34, 376, 3155, 9152, 10815, 4684, 475, 609, 5717, 19311, 28418, 17432, 3449, 89, 986, 10240, 39520
OFFSET
1,2
COMMENTS
Row n contains ceiling(n/2) terms.
Row sums are the odd-indexed Fibonacci numbers (A001519).
T(n,0) = Fibonacci(n+2) - 1 = A000071(n+2).
Sum_{k>=0} k*T(n,k) = A121532(n).
LINKS
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 = G(t,z) = z(1 - 2tz^2 - tz^3)(1-tz^2)/((1 - z - tz^2)(1 - z - z^2 - 3tz^2 - tz^3 + t^2*z^4)).
EXAMPLE
T(5,2)=2 because we have UU/UU/UDDDDD and UU/UDDU/UDDD, where U=(1,1) and D=(1,-1) (the double rises at an even level are indicated by a /).
Triangle starts:
1;
2;
4, 1;
7, 6;
12, 20, 2;
20, 51, 18;
33, 115, 80, 5;
MAPLE
G:=z*(1-2*t*z^2-t*z^3)*(1-t*z^2)/(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..ceil(n/2)-1) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 05 2006
STATUS
approved