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A191310 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k up-steps starting at level 0.

%I A191310 #15 Jul 17 2017 02:18:06
%S A191310 1,1,1,1,1,2,1,4,1,1,6,3,1,10,8,1,1,14,16,4,1,23,32,13,1,1,32,56,32,5,
%T A191310 1,55,102,74,19,1,1,78,170,152,55,6,1,143,302,307,144,26,1,1,208,498,
%U A191310 580,336,86,7,1,405,890,1102,748,251,34,1,1,602,1478,2004,1564,652,126,8,1,1228,2691,3714,3200,1587,405,43,1
%N A191310 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k up-steps starting at level 0.
%C A191310 Row n has 1 + floor(n/2) entries.
%C A191310 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191310 Sum_{k>=0} k*T(n,k) = A093387(n+1).
%F A191310 G.f.: G(t,z) = 2/(2-2*z-t*(1-sqrt(1-4*z^2))).
%e A191310 T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
%e A191310 Triangle starts:
%e A191310   1;
%e A191310   1;
%e A191310   1,  1;
%e A191310   1,  2;
%e A191310   1,  4,  1;
%e A191310   1,  6,  3;
%e A191310   1, 10,  8,  1;
%e A191310   1, 14, 16,  4;
%e A191310   1, 23, 32, 13,  1;
%p A191310 G := 2/(2-2*z-t*(1-sqrt(1-4*z^2))): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A191310 Cf. A001405, A093387.
%K A191310 nonn,tabf
%O A191310 0,6
%A A191310 _Emeric Deutsch_, May 30 2011