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 A191310 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k up-steps starting at level 0. 1
 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 3, 1, 10, 8, 1, 1, 14, 16, 4, 1, 23, 32, 13, 1, 1, 32, 56, 32, 5, 1, 55, 102, 74, 19, 1, 1, 78, 170, 152, 55, 6, 1, 143, 302, 307, 144, 26, 1, 1, 208, 498, 580, 336, 86, 7, 1, 405, 890, 1102, 748, 251, 34, 1, 1, 602, 1478, 2004, 1564, 652, 126, 8, 1, 1228, 2691, 3714, 3200, 1587, 405, 43, 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). Sum_{k>=0} k*T(n,k) = A093387(n+1). LINKS FORMULA G.f.: G(t,z) = 2/(2-2*z-t*(1-sqrt(1-4*z^2))). 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; 1, 4, 1; 1, 6, 3; 1, 10, 8, 1; 1, 14, 16, 4; 1, 23, 32, 13, 1; MAPLE G := 2/(2-2*z-t*(1-sqrt(1-4*z^2))): Gser := simplify(series(G, z = 0, 20)): 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, A093387. Sequence in context: A130313 A247073 A124428 * A124845 A191392 A127625 Adjacent sequences: A191307 A191308 A191309 * A191311 A191312 A191313 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, May 30 2011 STATUS approved

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Last modified April 2 01:40 EDT 2023. Contains 361723 sequences. (Running on oeis4.)