

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


2



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
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OFFSET

1,5


COMMENTS

Row n contains 1+floor(n/2) terms. Row sums are the oddsubscripted Fibonacci numbers (A001519). T(2n,n)=Fibonacci(2n1) (A001519). Sum(k*T(n,k), k>=0)=A121530(n).


LINKS

Table of n, a(n) for n=1..58.
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and qFibonacci numbers, Discrete Math., 170, 1997, 211217.


FORMULA

G.f.: G(t,z)=z(1tz^2)(1z+tzz^2tz^2t^2*z^3)/[(1ztz^2)(1zz^23tz^2tz^3+t^2*z^4)].


EXAMPLE

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;


MAPLE

G:=z*(1t*z^2)*(1z+t*zz^2t*z^2t^2*z^3)/(1zt*z^2)/(1zz^23*t*z^2t*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


CROSSREFS

Cf. A001519, A121530, A121531, A054142.
Sequence in context: A059926 A138775 A209385 * A304429 A006370 A262370
Adjacent sequences: A121526 A121527 A121528 * A121530 A121531 A121532


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Aug 05 2006


STATUS

approved



