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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
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
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
Sequence in context: A372505 A322338 A158971 * A366073 A362412 A376657
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 02 2006
STATUS
approved