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A191306 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 height of first peak equal to k.

%I A191306 #12 Jul 17 2017 02:17:57
%S A191306 1,1,1,1,1,2,1,4,1,1,7,2,1,13,5,1,1,23,9,2,1,43,19,6,1,1,78,34,11,2,1,
%T A191306 148,69,26,7,1,1,274,125,47,13,2,1,526,251,103,34,8,1,1,988,461,187,
%U A191306 62,15,2,1,1912,923,397,146,43,9,1,1,3628,1715,727,266,79,17,2,1,7060,3431,1519,596,199,53,10,1
%N A191306 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 height of first peak equal to k.
%C A191306 Row n has 1 + floor(n/2) entries.
%C A191306 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191306 T(n,1) = A036256(n-2).
%C A191306 Sum_{k>=0} k*T(n,k) = A191307(n).
%F A191306 G.f.: G=G(t,z) satisfies G = 1+z*G + t*z^2*g/(1-t*z^2*C), where C=1+z^2*C^2 and g=2/(1-2*z+sqrt(1-4*z^2)).
%e A191306 T(5,2)=2 because we have UUDDH and HUUDD, where U=(1,1), D=(1,-1), H=(1,0).
%e A191306 Triangle starts:
%e A191306   1;
%e A191306   1;
%e A191306   1,  1;
%e A191306   1,  2;
%e A191306   1,  4,  1;
%e A191306   1,  7,  2;
%e A191306   1, 13,  5,  1;
%e A191306   1, 23,  9,  2;
%e A191306   1, 43, 19,  6,  1;
%p A191306 C := ((1-sqrt(1-4*z^2))*1/2)/z^2: g := 2/(1-2*z+sqrt(1-4*z^2)): G := (1-t*z^2*C+t*z^2*g)/((1-t*z^2*C)*(1-z)): 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 A191306 Cf. A001405, A036256, A191307.
%K A191306 nonn,tabf
%O A191306 0,6
%A A191306 _Emeric Deutsch_, May 30 2011