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A191397 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)).

%I A191397 #11 Jul 17 2017 02:16:35
%S A191397 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 A191397 956,726,34,1948,1380,104,3214,2970,250,1,6591,5616,659,4,10959,11829,
%U A191397 1502,20,22609,22300,3647,64,37833,46306,8019,220,78486,87154,18495,620,1,132037,179222,39648,1804,5
%N A191397 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 A191397 Row n has 1+floor(n/5) entries.
%C A191397 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191397 T(n,0) = A191398(n).
%C A191397 Sum_{k>=0} k*T(n,k) = A191389(n-1).
%F A191397 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 A191397 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 A191397 Triangle starts:
%e A191397    1;
%e A191397    1;
%e A191397    2;
%e A191397    3;
%e A191397    6;
%e A191397    9,  1;
%e A191397   18,  2;
%e A191397   28,  7;
%e A191397   56, 14;
%p A191397 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 A191397 Cf. A001405, A191389, A191398.
%K A191397 nonn,tabf
%O A191397 0,3
%A A191397 _Emeric Deutsch_, Jun 04 2011