A191397 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) having k DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

1, 1, 2, 3, 6, 9, 1, 18, 2, 28, 7, 56, 14, 89, 37, 179, 72, 1, 289, 170, 3, 585, 326, 13, 956, 726, 34, 1948, 1380, 104, 3214, 2970, 250, 1, 6591, 5616, 659, 4, 10959, 11829, 1502, 20, 22609, 22300, 3647, 64, 37833, 46306, 8019, 220, 78486, 87154, 18495, 620, 1, 132037, 179222, 39648, 1804, 5

Offset

0,3

Comments

Row n has 1+floor(n/5) entries.

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

T(n,0) = A191398(n).

Sum_{k>=0} k*T(n,k) = A191389(n-1).

Links

Table of n, a(n) for n=0..59.

Formula

G.f.: G(t,z) = 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)).

Example

T(6,1)=2 because we have HU(DHU)D and U(DHU)DH, where U=(1,1), D=(1,-1), H=(1,0) (the DHU's are shown between parentheses).
Triangle starts:
   1;
   1;
   2;
   3;
   6;
   9,  1;
  18,  2;
  28,  7;
  56, 14;

Maple

G := 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 21 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 21 do seq(coeff(P[n], t, k), k = 0 .. floor((1/5)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A001405, A191389, A191398.

Sequence in context: A199790 A088329 A193079 * A358764 A361133 A361534

Adjacent sequences: A191394 A191395 A191396 * A191398 A191399 A191400

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 04 2011

Status

approved