%I A145895
%S A145895 1,1,0,1,1,2,1,0,1,2,4,2,4,1,0,1,4,8,10,8,4,6,1,0,1,8,20,26,24,25,12,7,
%T A145895 8,1,0,1,17,48,70,84,70,54,47,16,11,10,1,0,1,37,116,197,244,241,224,141,
%U A145895 104,76,20,16,12,1,0,1,82,286,535,728,816,734,609,472,246,180,112,24,22
%N A145895 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength 
               n and having k UDU and DUD's (here U=(1,1), D=(1,-1); 0<=k<=2n-2).
%C A145895 Row n contains 2n-1 entries (n>=1).
%C A145895 Row sums are the Catalan numbers (A000108).
%C A145895 T(n,0)=A004148(n-1) (the 2ndary structure numbers).
%C A145895 Sum(k*T(n,k),k=0..2n-2)=2*binom(2n-2,n-2)=2*A001791(n-1).
%F A145895 G.f. = c(z/[1+(1-t^2)z+(1-t)^2*z^2]), where c(z)=[1-sqrt(1-4z)](2z) is 
               the g.f. of the Catalan numbers (A000108).
%F A145895 The trivariate g.f., with z marking semilength, t marking number of UDU's 
               and s marking number of DUD's is c(z/[1+(1-ts)z+(1-t)(1-s)z^2]), 
               where c(z)=[1-sqrt(1-4z)](2z) is the g.f. of the Catalan numbers 
               (A000108).
%e A145895 T(4,3)=4 because we have UDUDUUDD, UUDDUDUD, UDUUDUDD and UUDUDDUD.
%e A145895 Triangle starts:
%e A145895 1;
%e A145895 1;
%e A145895 1,0,1;
%e A145895 1,2,1,0,1;
%e A145895 2,4,2,4,1,0,1;
%e A145895 4,8,10,8,4,6,1,0,1;
%p A145895 c := proc (z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end 
               proc: G := c(z/(1+(1-t^2)*z+(1-t)^2*z^2)): Gser := simplify(series(G, 
               z = 0, 12)): for n from 0 to 9 do P[n] := sort(coeff(Gser, z, n)) 
               end do: 1; for n to 9 do seq(coeff(P[n], t, k), k = 0 .. 2*n-2) end 
               do; # yields sequence in triangular form
%Y A145895 A000108, A004148, A001791
%Y A145895 Sequence in context: A022958 A023444 A136868 this_sequence A114503 A103528 
               A138352
%Y A145895 Adjacent sequences: A145892 A145893 A145894 this_sequence A145896 A145897 
               A145898
%K A145895 nonn,tabf
%O A145895 0,6
%A A145895 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2008