%I #11 Jul 17 2017 02:16:35

%S 1,1,2,3,6,9,1,18,2,28,7,56,14,89,37,179,72,1,289,170,3,585,326,13,

%T 956,726,34,1948,1380,104,3214,2970,250,1,6591,5616,659,4,10959,11829,

%U 1502,20,22609,22300,3647,64,37833,46306,8019,220,78486,87154,18495,620,1,132037,179222,39648,1804,5

%N Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having k DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

%C Row n has 1+floor(n/5) entries.

%C Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

%C T(n,0) = A191398(n).

%C Sum_{k>=0} k*T(n,k) = A191389(n-1).

%F G.f.: G(t,z) = 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)).

%e T(6,1)=2 because we have HU(DHU)D and U(DHU)DH, where U=(1,1), D=(1,-1), H=(1,0) (the DHU's are shown between parentheses).

%e Triangle starts:

%e 1;

%e 1;

%e 2;

%e 3;

%e 6;

%e 9, 1;

%e 18, 2;

%e 28, 7;

%e 56, 14;

%p G := 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 21 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 21 do seq(coeff(P[n], t, k), k = 0 .. floor((1/5)*n)) end do; # yields sequence in triangular form

%Y Cf. A001405, A191389, A191398.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Jun 04 2011