A191518 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of semilength n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having k UUU's (U=(1,1)).

1, 1, 2, 3, 6, 10, 19, 1, 33, 2, 62, 7, 1, 110, 14, 2, 205, 38, 8, 1, 368, 76, 16, 2, 683, 181, 50, 9, 1, 1235, 360, 101, 18, 2, 2286, 801, 270, 64, 10, 1, 4153, 1584, 546, 130, 20, 2, 7674, 3377, 1340, 387, 80, 11, 1, 13986, 6640, 2707, 790, 163, 22, 2, 25813, 13760, 6272, 2128, 536, 98, 12, 1

Offset

0,3

Comments

Row n (n>=4) contains floor(n/2)-1 entries.

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

T(n,0) = A191519(n).

Sum_{k>=0} k*T(n,k) = A191520(n).

Links

Alois P. Heinz, Rows n = 0..220, flattened

Formula

G.f.: G=G(t,z) satisfies aG^2 + bG -1 = 0, where a=z(1-z-z^2-z^3-tz+tz^2+tz^3), and b=1-2z-z^2+tz^2.

Example

T(7,1) = 2 because we have UUUDDDH and HUUUDDD, where U=(1,1), H=(1,0), and D=(1,-1).
Triangle starts:
   1;
   1;
   2;
   3;
   6;
  10;
  19, 1;
  33, 2;
  62, 7, 1;

Maple

a := z*(1-z-z^2-z^3-t*z+t*z^2+t*z^3): b := 1-2*z-z^2+t*z^2: G := RootOf(a*g^2+b*g-1 = 0, g): Gser := simplify(series(G, z = 0, 21)): for n from 0 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; 2; 3; for n from 4 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(b(x-1, y+1, min(t+1, 3))*
      `if`(t=3, z, 1) +b(x-1, y-1, 1)+ `if`(y=0, b(x-1, 0, 1), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jun 02 2014

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, Min[t+1, 3]]*If[t == 3, z, 1] + b[x-1, y-1, 1] + If[y == 0, b[x-1, 0, 1], 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

Crossrefs

Cf. A001405, A191519, A191520.

Sequence in context: A240802 A339808 A089985 * A217382 A378672 A244742

Adjacent sequences: A191515 A191516 A191517 * A191519 A191520 A191521

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 07 2011

Status

approved