%I A114515
%S A114515 0,0,1,3,12,45,171,651,2488,9540,36690,141482,546864,2118207,8219967,
%T A114515 31952115,124389552,484908408,1892657934,7395597354,28928182440,
%U A114515 113260606074,443827115886,1740592240638,6831289801872,26829201570600
%N A114515 Number of peaks in all hill-free Dyck paths of semilength n (a Dyck path 
               is hill-free if it has no peaks at level 1).
%C A114515 a(n)=Sum(k*A100754(n,k), k=0..n-1).
%F A114515 G.f.=z^2*C/[(1-zC+z)^2*(1-2zC)}, where C=[1-sqrt(1-4z)]/(2z) is the Catalan 
               function.
%e A114515 a(3)=3 because in the two hill-free Dyck paths of semilength 3, namely 
               U(UD)(UD)D and UU(UD)DD, we have alltogether 3 peaks (shown between 
               parantheses).
%p A114515 C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z)^2*z^2*C/(1-2*z*C): Gser:=series(G,
               z=0,32): 0, seq(coeff(Gser,z^n),n=1..28);
%Y A114515 Cf. A100754.
%Y A114515 Sequence in context: A128593 A085481 A030195 this_sequence A151162 A094547 
               A026559
%Y A114515 Adjacent sequences: A114512 A114513 A114514 this_sequence A114516 A114517 
               A114518
%K A114515 nonn
%O A114515 0,4
%A A114515 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 04 2005