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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.
1

%I #12 Jul 17 2017 02:17:57

%S 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 148,69,26,7,1,1,274,125,47,13,2,1,526,251,103,34,8,1,1,988,461,187,

%U 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 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 Row n has 1 + floor(n/2) entries.

%C Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

%C T(n,1) = A036256(n-2).

%C Sum_{k>=0} k*T(n,k) = A191307(n).

%F 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 T(5,2)=2 because we have UUDDH and HUUDD, where U=(1,1), D=(1,-1), H=(1,0).

%e Triangle starts:

%e 1;

%e 1;

%e 1, 1;

%e 1, 2;

%e 1, 4, 1;

%e 1, 7, 2;

%e 1, 13, 5, 1;

%e 1, 23, 9, 2;

%e 1, 43, 19, 6, 1;

%p 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 Cf. A001405, A036256, A191307.

%K nonn,tabf

%O 0,6

%A _Emeric Deutsch_, May 30 2011