A121467 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and area k (n>=1, k>=1).

1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 3, 0, 2, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 4, 0, 4, 0, 5, 0, 5, 0, 4, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 7, 0, 8, 0, 10, 0, 8, 0, 11, 0, 9, 0, 7, 0, 6, 0, 6, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 11, 0, 13

Offset

1,10

Comments

Row n has n^2 terms. Row sums are the odd-subscripted Fibonacci numbers (A001519). Sum(k*T(n,k),k=1..n^2)=A061648(n).

Links

Table of n, a(n) for n=1..104.

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(q,x)=Sum(F[n]x^n, n>=0), where the q-analogs F[n] of the Fibonacci numbers are defined by F[0]=0, F[1]=q, F[n+2]=F[n+1]q^(2n+3)+Sum(F[n-k+1]q^((k+1)^2),k=0..n).

Example

T(3,5)=2 because we have UDUUDD and UUDDUD, where U=(1,1) and D(1,-1).
Triangle starts:
1;
0,1,0,1;
0,0,1,0,2,0,1,0,1;
0,0,0,1,0,3,0,2,0,3,0,2,0,1,0,1;

Maple

P[1]:=q: for n from 2 to 10 do P[n]:=sort(expand(q^(2*n-1)*P[n-1]+sum(q^((k+1)^2)*P[n-k-1], k=0..n-2))) od: for n from 1 to 7 do seq(coeff(P[n], q, j), j=1..n^2) od; # yields sequence in triangular form

Crossrefs

Cf. A001519, A061648.

Sequence in context: A369934 A322338 A158971 * A366073 A362412 A071325

Adjacent sequences: A121464 A121465 A121466 * A121468 A121469 A121470

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 02 2006

Status

approved