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A191390 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k horizontal segments.

%I A191390 #14 Jul 17 2017 02:16:30
%S A191390 1,0,1,1,1,0,3,2,3,1,0,8,2,5,8,7,0,22,12,1,14,22,31,3,0,64,50,12,42,
%T A191390 64,117,28,1,0,196,184,78,4,132,196,416,162,18,0,625,648,390,52,1,429,
%U A191390 625,1452,762,159,5,0,2055,2256,1707,392,25,1430,2055,5062,3225,1012,85,1,0,6917,7868,6954,2280,285,6
%N A191390 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k horizontal segments.
%C A191390 A dispersed Dyck paths of length n is a Motzkin paths of length n with no (1,0)-steps at positive heights. A horizontal segment is a maximal sequence of consecutive (1,0)-steps.
%C A191390 Row n has 1 + ceiling(n/3) entries.
%C A191390 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191390 T(2n,0) = A000108(n) (the Catalan numbers); T(2n+1,0) = 0.
%C A191390 T(2n-1,1) = T(2n,1) = A014138(n) (partial sums of Catalan numbers).
%C A191390 Sum_{k>=0} k*T(n,k) = A191391(n).
%F A191390 G.f.: G(t,z) = (2*(1-z+t*z))/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)).
%e A191390 T(5,2)=2 because we have (HH)UD(H) and (H)UD(HH), where U=(1,1), D=(1,-1), H=(1,0) (the horizontal segments are shown between parentheses).
%e A191390 Triangle starts:
%e A191390   1;
%e A191390   0,  1;
%e A191390   1,  1;
%e A191390   0,  3;
%e A191390   2,  3,  1;
%e A191390   0,  8,  2;
%e A191390   5,  8,  7;
%e A191390   0, 22, 12,  1;
%p A191390 G := (2*(1-z+t*z))/(1-z-t*z+(1-z+t*z)*sqrt(1-4*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: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. ceil((1/3)*n)) end do; # yields sequence in triangular form
%Y A191390 Cf. A000108, A001405, A014138, A191391.
%K A191390 nonn,tabf
%O A191390 0,7
%A A191390 _Emeric Deutsch_, Jun 03 2011