%I #3 Mar 30 2012 17:36:07
%N Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at level 2; 0<= k<=n-1, n>=2 (a Dyck path is said to be hill-free if it has no peaks at level 1).
%C Row n has n terms (n>=2). Row sums yield the Fine numbers (A000957). T(n,0)=A114627(n-3). Sum(kT(n,k),k=0..n-1)=A114495(n).
%F G.f.=(1+z-tz-zC)/[1+z+z^2-tz-tz^2-z(1+z)C], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e T(5,2)=3 because we have U(UD)(UD)UUDDD, UUUDD(UD)(UD)D and U(UD)UUDD(UD)D, where U=(1,1), D=(1,-1) (the peaks at level 2 are shown between parentheses).
%e Triangle begins:
%p C:=(1-sqrt(1-4*z))/2/z: G:=(1+z-t*z-z*C)/(1+z+z^2-t*z-t*z^2-z*(1+z)*C): 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) od; # yields sequence in triangular form
%Y Cf. A000957, A114627, A114495.
%A _Emeric Deutsch_, Dec 18 2005