A191320 - OEIS

%I #11 Jul 17 2017 01:05:00

%S 1,1,2,2,1,4,2,4,6,9,10,1,9,23,3,23,36,11,23,77,25,1,65,118,65,4,65,

%T 249,131,17,197,380,298,48,1,197,808,566,140,5,626,1236,1210,336,24,

%U 626,2665,2230,833,80,1,2056,4094,4627,1828,259,6,2056,8955,8401,4155,711,32,6918,13816,17192,8648,1923,122,1

%N 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) with k HUs, where U=(1,1) and H=(1,0).

%C Row n has 1 + floor(n/3) entries.

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

%C T(2*n,0) = T(2*n+1,0) = A014137(n) (partial sums of the Catalan numbers).

%C Sum_{k>=0}k*T(n,k) = A093387(n).

%F G.f.: G(t,z) = 2/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)).

%e T(7,2)=3 because we have HHUDHUD, HUDHHUD, and HUDHUDH, where U=(1,1), D=(1,-1), H=(1,0).

%e Triangle starts:

%e    1;

%e    1;

%e    2;

%e    2,  1;

%e    4,  2;

%e    4,  6;

%e    9, 10,  1;

%e    9, 23,  3;

%e   23, 36, 11;

%p G := 2/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

%Y Cf. A001405, A014137, A093387.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Jun 01 2011