A132893 - OEIS

A132893

Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having k peaks (i.e., UDs), 0 <= k <= floor(n/2).

1, 2, 4, 1, 9, 4, 21, 13, 1, 50, 40, 6, 121, 118, 27, 1, 296, 340, 106, 8, 730, 965, 381, 46, 1, 1812, 2708, 1296, 220, 10, 4521, 7535, 4241, 935, 70, 1, 11328, 20828, 13482, 3676, 395, 12, 28485, 57266, 41916, 13658, 1940, 99, 1

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OFFSET

0,2

COMMENTS

Row n has 1 + floor(n/2) terms.

Row sums yield A005773.

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.

FORMULA

T(n,0) = A091964(n).

Sum_{k=0..floor(n/2)} k*T(n,k) = A132894(n-1).

G.f.: G = G(t,z) satisfies z(1 - 3z + z^2 - tz^2)G^2 + (1 - 3z + z^2 - tz^2)G - 1 = 0 (see the Maple program for the explicit expression of G).

EXAMPLE

T(3,1)=4 because we have HUD, UDH, UDU and UUD.

Triangle starts:

4, 1;

9, 4;

21, 13, 1;

50, 40, 6;

121, 118, 27, 1;

MAPLE

G:=((-1+3*z-z^2+t*z^2+sqrt((1+z+z^2-t*z^2)*(1-3*z+z^2-t*z^2)))*1/2)/(z*(1-3*z+z^2-t*z^2)): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(x, y, t) option remember; expand(`if`(y<0, 0,

`if`(x=0, 1, b(x-1, y, 1)+b(x-1, y-1, 1)*t+b(x-1, y+1, z))))

end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, 1)):

seq(T(n), n=0..15); # Alois P. Heinz, Feb 01 2019

MATHEMATICA

b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0, 0, If[x == 0, 1, b[x - 1, y, 1] + b[x - 1, y - 1, 1]*t + b[x - 1, y + 1, z]]]];

T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]] @ b[n, 0, 1];

T /@ Range[0, 15] // Flatten (* Jean-François Alcover, Oct 06 2019, after Alois P. Heinz *)

CROSSREFS

Cf. A005773, A091964, A132894.