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 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 (list; graph; refs; listen; history; text; internal format)
 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). T(n,0) = A191385(n). Sum_{k>=0} k*T(n,k) = A191386(n). LINKS Table of n, a(n) for n=0..80. FORMULA 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)). 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-(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 CROSSREFS Cf. A001405, A191385, A191386. Sequence in context: A331886 A205456 A080045 * A191305 A227287 A289236 Adjacent sequences: A191381 A191382 A191383 * A191385 A191386 A191387 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 01 2011 STATUS approved

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Last modified August 6 15:42 EDT 2024. Contains 374974 sequences. (Running on oeis4.)