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A191387 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k valleys at level 0.

%I A191387 #18 Sep 24 2021 08:31:12
%S A191387 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 A191387 214,164,66,16,2,393,330,149,43,8,1,682,618,301,94,19,2,1267,1225,648,
%U A191387 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 A191387 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k valleys at level 0.
%C A191387 A dispersed Dyck paths of length n is a Motzkin path of length n with no (1,0) steps at positive heights.
%C A191387 Row n >=2 has floor(n/2) entries.
%C A191387 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191387 T(n,0) = A191388(n).
%C A191387 Sum_{k>=0} k*T(n,k) = A191389(n).
%H A191387 Alois P. Heinz, <a href="/A191387/b191387.txt">Rows n = 0..200, flattened</a>
%F A191387 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 A191387 T(5,1)=2 because we have HUDUD and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
%e A191387 Triangle starts:
%e A191387    1;
%e A191387    1;
%e A191387    2;
%e A191387    3;
%e A191387    5,  1;
%e A191387    8,  2;
%e A191387   14,  5,  1;
%e A191387   23, 10,  2;
%e A191387   41, 22,  6,  1;
%e A191387   ...
%p A191387 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 A191387 Cf. A001405, A191388, A191389.
%K A191387 nonn,tabf
%O A191387 0,3
%A A191387 _Emeric Deutsch_, Jun 02 2011