

A191384


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 ascents of length 1. An ascent is a maximal sequence of consecutive (1,1)steps.


2



1, 1, 1, 1, 1, 2, 2, 3, 1, 3, 4, 3, 5, 8, 6, 1, 7, 14, 10, 4, 12, 26, 21, 10, 1, 18, 42, 41, 20, 5, 31, 77, 83, 45, 15, 1, 47, 128, 150, 96, 35, 6, 81, 234, 293, 209, 85, 21, 1, 125, 388, 530, 414, 196, 56, 7, 216, 704, 1023, 858, 455, 147, 28, 1, 337, 1172, 1828, 1668, 974, 364, 84, 8, 583, 2119, 3479, 3385, 2133, 896, 238, 36, 1
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OFFSET

0,6


COMMENTS

Row n has 1 + floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).


LINKS



FORMULA

G.f.: G(t,z) = (t*z^2  (1z)^2 + sqrt((1+z^2t*z^2)*(13*z^2t*z^2)))/(2*z*(12*z+z^2z^3t*z^2+t*z^3)).


EXAMPLE

T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,1), H=(1,0).
Triangle starts:
1;
1;
1, 1;
1, 2;
2, 3, 1;
3, 4, 3;
5, 8, 6, 1;
7, 14, 10, 4;


MAPLE

G := ((t*z^2(1z)^2+sqrt((1+z^2t*z^2)*(13*z^2t*z^2)))*1/2)/(z*(12*z+z^2z^3t*z^2+t*z^3)): Gser := simplify(series(G, z = 0, 19)): 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


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



