%I
%S 1,0,1,1,0,1,1,3,0,1,3,3,6,0,1,5,14,5,9,0,1,12,22,35,7,12,0,1,22,68,
%T 53,65,9,15,0,1,49,127,203,97,104,11,18,0,1,94,329,390,444,153,152,13,
%U 21,0,1,201,664,1157,873,816,221,209,15,24,0,1,396,1576,2456,2925,1627,1345
%N Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at odd level (0<=k<=n).
%C Row sums are the oddsubscripted Fibonacci numbers (A001519). T(n,0)=A121482(n). Sum(k*T(n,k),k=0..n)=A121483(n).
%H E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012365X(97)827781">Nondecreasing Dyck paths and qFibonacci numbers</a>, Discrete Math., 170, 1997, 211217.
%F G.f.: G(t,z) = (tz^2+z^2+z1)(tz^3+2z^21)/(14z^2ztz+2tz^4+4z^4z^6+2z^3+tz^3+t^2*z^3).
%e T(3,1)=3 because we have UDUUDUDD, UUDUUDDD and UUUDDUDD, where U=(1,1) and D=(1,1) (the peaks at odd level are shown by a ; the Dyck path UUDUDDUD has 1 peak at odd level but it is not nondecreasing).
%e Triangle starts:
%e 1;
%e 0,1;
%e 1,0,1;
%e 1,3,0,1;
%e 3,3,6,0,1;
%e 5,14,5,9,0,1;
%p G:=(t*z^2+z^2+z1)*(t*z^3+2*z^21)/(14*z^2zt*z+2*z^4*t+4*z^4z^6+2*z^3+t*z^3+z^3*t^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y Cf. A001519, A121482, A121483, A121484.
%K nonn,tabl
%O 0,8
%A _Emeric Deutsch_, Aug 02 2006
