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, 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

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

Table of n, a(n) for n=0..80.

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