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A191305 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 hills (i.e., peaks at height 1).

%I A191305 #10 Jul 17 2017 03:53:56
%S A191305 1,1,1,1,1,2,2,3,1,3,4,3,6,7,6,1,9,12,10,4,18,23,18,10,1,28,40,33,20,
%T A191305 5,57,76,64,39,15,1,91,134,120,76,35,6,187,257,231,152,75,21,1,304,
%U A191305 460,433,300,156,56,7,629,888,834,595,325,132,28,1,1037,1606,1572,1164,670,294,84,8,2157,3115,3035,2292,1375,642,217,36,1
%N A191305 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 hills (i.e., peaks at height 1).
%C A191305 Row n has 1 + floor(n/2) entries.
%C A191305 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191305 Sum_{k>=0}k*T(n,k) = A045621(n-2).
%F A191305 G.f.: G=G(t,z) satisfies G = 1+z*G + z^2*G(C-1+t), where C=1+z^2*C^2 (and G=2/(1-2*z+2*z^2-2*t*z^2+sqrt(1-4*z^2)), see Maple program).
%e A191305 T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
%e A191305 Triangle starts:
%e A191305   1;
%e A191305   1;
%e A191305   1,  1;
%e A191305   1,  2;
%e A191305   2,  3,  1;
%e A191305   3,  4,  3;
%e A191305   6,  7,  6,  1;
%e A191305   9, 12, 10,  4;
%p A191305 G := 2/(1-2*z+2*z^2-2*t*z^2+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 A191305 Cf. A001405, A045621.
%K A191305 nonn,tabf
%O A191305 0,6
%A A191305 _Emeric Deutsch_, May 30 2011