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A114586 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at odd levels (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.

%I A114586 #4 Mar 30 2012 17:36:07
%S A114586 1,1,1,3,2,1,6,8,3,1,15,22,15,4,1,36,68,52,24,5,1,91,198,191,100,35,6,
%T A114586 1,232,586,651,425,170,48,7,1,603,1718,2203,1656,820,266,63,8,1,1585,
%U A114586 5047,7285,6299,3591,1435,392,80,9,1,4213,14808,23832,23164,15155,6972,2338
%N A114586 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at odd levels (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.
%C A114586 Row sums are the Fine numbers (A000957). Column 0 yield the Riordan numbers (A005043). Sum(k*T(n,k),k=0..n-2)=A114587(n).
%F A114586 G.f.=G-1, where G=G(t, z) satisfies z(1+t+z)G^2-(1+z+tz)G+1=0.
%e A114586 T(5,2)=3 because we have UU(UD)DU(UD)DD, UUDU(UD)(UD)DD and UU(UD)(UD)DUDD, where U=(1,1), D=(1,-1) (the peaks at odd levels are shown between parentheses).
%e A114586 Triangle begins:
%e A114586 1;
%e A114586 1,1;
%e A114586 3,2,1;
%e A114586 6,8,3,1;
%e A114586 15,22,15,4,1;
%p A114586 G:=(t*z+z+1-sqrt(z^2*t^2+2*z^2*t-2*z*t-3*z^2-2*z+1))/2/z/(1+t+z)-1: Gser:=simplify(series(G,z=0,15)): for n from 2 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 2 to 12 do seq(coeff(t*P[n],t^j),j=1..n-1) od; # yields sequence in triangular form
%Y A114586 Cf. A000957, A005043, A114587, A114588, A100754.
%K A114586 nonn,tabl
%O A114586 2,4
%A A114586 _Emeric Deutsch_, Dec 11 2005