A121522 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k up steps starting at an even level (1<=k<=n). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 8, 15, 9, 1, 1, 11, 30, 34, 12, 1, 1, 14, 55, 85, 62, 15, 1, 1, 17, 89, 185, 200, 99, 18, 1, 1, 20, 132, 365, 510, 402, 145, 21, 1, 1, 23, 184, 650, 1160, 1220, 718, 200, 24, 1, 1, 26, 245, 1067, 2400, 3155, 2585, 1175, 264, 27, 1, 1, 29, 315

Offset

1,5

Comments

Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,k)=A121524(n,n-k), i.e. triangle is mirror image of A121524. Sum(k*T(n,k), k=1..n)=A121523(n).

Links

Table of n, a(n) for n=1..69.

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(1-tz^2)(1-2tz^2-tz^3)/(1-z-tz-4tz^2+2tz^3+2t^2*z^3+6t^2*z^4-t^3*z^6).

Example

T(4,2)=5 because we have (U)D(U)UDUDD, (U)UDD(U)UDD, (U)UDU(U)DDD, (U)U(U)DDUDD and (U)U(U)UDDDD, where U=(1,1) and D=(1,-1) (the up steps starting at an even level are shown between parentheses; UUDUDDUD does not qualify because it is not nondecreasing).
Triangle starts:
1;
1,1;
1,3,1;
1,5,6,1;
1,8,15,9,1;
1,11,30,34,12,1;

Maple

g:=t*z*(1-t*z^2)*(1-2*t*z^2-t*z^3)/(1-z-t*z-4*t*z^2+2*t*z^3+2*t^2*z^3+6*t^2*z^4-t^3*z^6): gser:=simplify(series(g, z=0, 17)): for n from 1 to 12 do P[n]:=sort(expand(coeff(gser, z, n))) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form

Crossrefs

Cf. A001519, A121523, A121524.

Sequence in context: A076756 A114172 A271942 * A294582 A294589 A204027

Adjacent sequences: A121519 A121520 A121521 * A121523 A121524 A121525

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 05 2006

Status

approved