%I A110190
%S A110190 0,1,5,24,116,568,2820,14184,72180,371112,1925380,10068728,53023860,
%T A110190 280969560,1497072132,8016213960,43114424308,232817773640,1261793848836,
%U A110190 6861179441880,37421756333172,204671007577464,1122275850740996
%N A110190 Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths 
               of length 2n (a Schroeder path of length 2n is a path from (0,0) 
               to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and 
               never going below the x-axis).
%C A110190 a(n)=sum(k*A110189(n,k), k=0..n).
%F A110190 G.f.=z(1-z-2zR+z^2+2z^2*R+z^2*R^2)/(1-3z-zR+z^2+z^2*R)^2, where R=1+zR+zR^2={1-z-sqrt(1-6z+z^2)]/
               (2z) is the g.f. for the large Schroeder numbers (A006318).
%e A110190 a(2)=5 because in the 6 (=A006318(2)) Schroeder paths of length 4, namely, 
               HH, HUD, UDH, UDUD, UHD, UUDD, all 5 H-steps are at levels 0 or 1.
%p A110190 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-2*z*R+z^2+2*z^2*R+z^2*R^2)/(1-3*z-z*R+z^2+z^2*R)^2: 
               Gser:=series(G,z=0,30): 0,seq(coeff(Gser,z^n),n=1..26);
%Y A110190 Cf. A006318, A110189.
%Y A110190 Sequence in context: A004254 A086347 A026707 this_sequence A026784 A017977 
               A017978
%Y A110190 Adjacent sequences: A110187 A110188 A110189 this_sequence A110191 A110192 
               A110193
%K A110190 nonn
%O A110190 0,3
%A A110190 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 15 2005