%I A118972
%S A118972 0,0,1,1,0,1,3,2,0,1,10,5,2,0,1,33,16,5,2,0,1,111,51,16,5,2,0,1,379,168,
%T A118972 51,16,5,2,0,1,1312,565,168,51,16,5,2,0,1,4596,1934,565,168,51,16,5,2,
               0,
%U A118972 1,16266,6716,1934,565,168,51,16,5,2,0,1,58082,23604,6716,1934,565,168
%N A118972 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of 
               semilength n and having length of first descent equal to k (1<=k<=n; 
               n>=1). A hill in a Dyck path is a peak at level 1.
%C A118972 Row sums are the Fine numbers (A000957). T(n,1)=A001558(n-3) for n>=3. 
               T(n,k)=A118973(n-k) for n>=k>=2. Sum(k*T(n,k),k=1..n)=A118974(n)
%D A118972 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 
               241, 241-265, 2001.
%F A118972 G:=tz^2*CF[C-(1-t)/(1-tz)], where F=[1-sqrt(1-4z)]/[z(3-sqrt(1-4z)] and 
               C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e A118972 T(5,2)=5 because we have uu(dd)uududd, uu(dd)uuuddd,uuu(dd)uuddd,uuu(dd)ududd 
               and uuuu(dd)uddd, where u=(1,1), d=(1,-1) (the first descents are 
               shown between parentheses).
%e A118972 Triangle starts:
%e A118972 0;
%e A118972 0,1;
%e A118972 1,0,1;
%e A118972 3,2,0,1;
%e A118972 10,5,2,0,1;
%e A118972 33,16,5,2,0,1;
%p A118972 F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: G:=t*z^2*C*F*(C-(1-t)/
               (1-t*z)): Gser:=simplify(series(G,z=0,15)): for n from 1 to 12 do 
               P[n]:=sort(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],
               t,j),j=1..n) od; # yields sequence in triangular form
%Y A118972 Cf. A000957, A001558, A118973, A118974.
%Y A118972 Sequence in context: A054654 A154477 A142071 this_sequence A145878 A112606 
               A108512
%Y A118972 Adjacent sequences: A118969 A118970 A118971 this_sequence A118973 A118974 
               A118975
%K A118972 nonn,tabl
%O A118972 1,7
%A A118972 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006