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%I #14 Jul 17 2017 02:16:30
%S 1,0,1,1,1,0,3,2,3,1,0,8,2,5,8,7,0,22,12,1,14,22,31,3,0,64,50,12,42,
%T 64,117,28,1,0,196,184,78,4,132,196,416,162,18,0,625,648,390,52,1,429,
%U 625,1452,762,159,5,0,2055,2256,1707,392,25,1430,2055,5062,3225,1012,85,1,0,6917,7868,6954,2280,285,6
%N Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k horizontal segments.
%C A dispersed Dyck paths of length n is a Motzkin paths of length n with no (1,0)-steps at positive heights. A horizontal segment is a maximal sequence of consecutive (1,0)-steps.
%C Row n has 1 + ceiling(n/3) entries.
%C Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C T(2n,0) = A000108(n) (the Catalan numbers); T(2n+1,0) = 0.
%C T(2n-1,1) = T(2n,1) = A014138(n) (partial sums of Catalan numbers).
%C Sum_{k>=0} k*T(n,k) = A191391(n).
%F G.f.: G(t,z) = (2*(1-z+t*z))/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)).
%e T(5,2)=2 because we have (HH)UD(H) and (H)UD(HH), where U=(1,1), D=(1,-1), H=(1,0) (the horizontal segments are shown between parentheses).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 1, 1;
%e 0, 3;
%e 2, 3, 1;
%e 0, 8, 2;
%e 5, 8, 7;
%e 0, 22, 12, 1;
%p G := (2*(1-z+t*z))/(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 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. ceil((1/3)*n)) end do; # yields sequence in triangular form
%Y Cf. A000108, A001405, A014138, A191391.
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Jun 03 2011
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