A191306 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 height of first peak equal to k.

1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 2, 1, 13, 5, 1, 1, 23, 9, 2, 1, 43, 19, 6, 1, 1, 78, 34, 11, 2, 1, 148, 69, 26, 7, 1, 1, 274, 125, 47, 13, 2, 1, 526, 251, 103, 34, 8, 1, 1, 988, 461, 187, 62, 15, 2, 1, 1912, 923, 397, 146, 43, 9, 1, 1, 3628, 1715, 727, 266, 79, 17, 2, 1, 7060, 3431, 1519, 596, 199, 53, 10, 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).

T(n,1) = A036256(n-2).

Sum_{k>=0} k*T(n,k) = A191307(n).

Links

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

Formula

G.f.: G=G(t,z) satisfies G = 1+z*G + t*z^2*g/(1-t*z^2*C), where C=1+z^2*C^2 and g=2/(1-2*z+sqrt(1-4*z^2)).

Example

T(5,2)=2 because we have UUDDH and HUUDD, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
  1;
  1;
  1,  1;
  1,  2;
  1,  4,  1;
  1,  7,  2;
  1, 13,  5,  1;
  1, 23,  9,  2;
  1, 43, 19,  6,  1;

Maple

C := ((1-sqrt(1-4*z^2))*1/2)/z^2: g := 2/(1-2*z+sqrt(1-4*z^2)): G := (1-t*z^2*C+t*z^2*g)/((1-t*z^2*C)*(1-z)): 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, A036256, A191307.

Sequence in context: A137633 A168533 A191314 * A191525 A066633 A238415

Adjacent sequences: A191303 A191304 A191305 * A191307 A191308 A191309

Keyword

nonn,tabf

Author

Emeric Deutsch, May 30 2011

Status

approved