%I #18 Sep 24 2021 08:31:12

%S 1,1,2,3,5,1,8,2,14,5,1,23,10,2,41,22,6,1,69,42,13,2,125,87,32,7,1,

%T 214,164,66,16,2,393,330,149,43,8,1,682,618,301,94,19,2,1267,1225,648,

%U 227,55,9,1,2223,2288,1290,484,126,22,2,4171,4498,2700,1100,322,68,10,1,7385,8396,5322,2300,718,162,25,2

%N Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k valleys at level 0.

%C A dispersed Dyck paths of length n is a Motzkin path of length n with no (1,0) steps at positive heights.

%C Row n >=2 has floor(n/2) entries.

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

%C T(n,0) = A191388(n).

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

%H Alois P. Heinz, <a href="/A191387/b191387.txt">Rows n = 0..200, flattened</a>

%F G.f.: G=G(t,z) is given by G = 1 + z*G + z^2*c*(t*(G-1-z*G) + 1 + z*G), where c = (1-sqrt(1-4*z^2))/(2*z^2).

%e T(5,1)=2 because we have HUDUD and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).

%e Triangle starts:

%e 1;

%e 1;

%e 2;

%e 3;

%e 5, 1;

%e 8, 2;

%e 14, 5, 1;

%e 23, 10, 2;

%e 41, 22, 6, 1;

%e ...

%p G := (1+z^2*c-t*z^2*c)/(1-z-z^3*c-t*z^2*c*(1-z)): c := ((1-sqrt(1-4*z^2))*1/2)/z^2: Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)-1) end do; # yields sequence in triangular form

%Y Cf. A001405, A191388, A191389.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Jun 02 2011