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%I #12 Jul 17 2017 02:16:20
%S 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,
%T 5,31,77,83,45,15,1,47,128,150,96,35,6,81,234,293,209,85,21,1,125,388,
%U 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
%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 ascents of length 1. An ascent is a maximal sequence of consecutive (1,1)-steps.
%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,0) = A191385(n).
%C Sum_{k>=0} k*T(n,k) = A191386(n).
%F G.f.: G(t,z) = (t*z^2 - (1-z)^2 + sqrt((1+z^2-t*z^2)*(1-3*z^2-t*z^2)))/(2*z*(1-2*z+z^2-z^3-t*z^2+t*z^3)).
%e T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
%e Triangle starts:
%e 1;
%e 1;
%e 1, 1;
%e 1, 2;
%e 2, 3, 1;
%e 3, 4, 3;
%e 5, 8, 6, 1;
%e 7, 14, 10, 4;
%p G := ((t*z^2-(1-z)^2+sqrt((1+z^2-t*z^2)*(1-3*z^2-t*z^2)))*1/2)/(z*(1-2*z+z^2-z^3-t*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
%Y Cf. A001405, A191385, A191386.
%K nonn,tabf
%O 0,6
%A _Emeric Deutsch_, Jun 01 2011
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