%I A116914
%S A116914 1,1,5,16,58,211,781,2920,11006,41746,159154,609324,2341060,9021559,
%T A116914 34855741,134972368,523689718,2035462990,7923732118,30889008112,
%U A116914 120566373676,471134916286,1842964183570,7216096752496,28279240308268
%N A116914 Number of UUDD's, where U=(1,1) and D=(1,-1), in all hill-free Dyck paths 
               of semilength n (a hill in a Dyck path is a peak at level 1).
%C A116914 a(n)=Sum(k*A105640(n,k), k=0..floor(n/2)).
%C A116914 Catalan transform of A034299. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jun 29 2009]
%D A116914 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 
               241 (2001), 241-265.
%F A116914 G.f.=z[1+5z-(1-z)sqrt(1-4z)]/[2(2+z)^2*sqrt(1-4z)].
%F A116914 a(n+2)=A126258(2*n,n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 
               13 2007
%e A116914 a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 
               4, namely UU(UUDD)DD, UUUDUDDD, UUD(UUDD)D, UUDUDUDD, U(UUDD)UDD 
               and (UUDD)(UUDD) (U=(1,1), D=(1,-1)) we have altogether 5 UUDD's 
               (shown between parentheses).
%p A116914 G:=z*(1+5*z-(1-z)*sqrt(1-4*z))/2/(2+z)^2/sqrt(1-4*z): Gser:=series(G,
               z=0,31): seq(coeff(Gser,z^n),n=2..28);
%Y A116914 Cf. A105640.
%Y A116914 Sequence in context: A153366 A057553 A006217 this_sequence A047103 A077235 
               A098347
%Y A116914 Adjacent sequences: A116911 A116912 A116913 this_sequence A116915 A116916 
               A116917
%K A116914 nonn
%O A116914 2,3
%A A116914 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006