A121467 - OEIS

%I #6 Mar 06 2018 11:37:30

%S 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,

%T 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,

%U 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

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

%C 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).

%H E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170, 1997, 211-217.

%F 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).

%e T(3,5)=2 because we have UDUUDD and UUDDUD, where U=(1,1) and D(1,-1).

%e Triangle starts:

%e 1;

%e 0,1,0,1;

%e 0,0,1,0,2,0,1,0,1;

%e 0,0,0,1,0,3,0,2,0,3,0,2,0,1,0,1;

%p 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

%Y Cf. A001519, A061648.

%K nonn,tabf

%O 1,10

%A _Emeric Deutsch_, Aug 02 2006