A097862 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and height k (n>=0, k>=0).

1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 5, 1, 31, 18, 1, 1, 63, 56, 7, 1, 127, 161, 33, 1, 1, 255, 441, 129, 9, 1, 511, 1170, 453, 52, 1, 1, 1023, 3036, 1485, 242, 11, 1, 2047, 7753, 4644, 990, 75, 1, 1, 4095, 19565, 14040, 3718, 403, 13, 1, 8191, 48930, 41392, 13145, 1872, 102

Offset: 0

Author

Emeric Deutsch, Sep 01 2004

Keywords

• nonn
• tabf

Comments

Row sums are the Motzkin numbers (A001006).

Examples

			Triangle begins:
  1;
  1;
  1,   1;
  1,   3;
  1,   7,    1;
  1,  15,    5;
  1,  31,   18,   1;
  1,  63,   56,   7;
  1, 127,  161,  33,  1;
  1, 255,  441, 129,  9;
  1, 511, 1170, 453, 52, 1;
  ...
Row n contains 1+floor(n/2) terms.
T(5,2) = 5 counts HUUDD, UUDDH, UUDHD, UHUDD and UUHDD, where U=(1,1), H=(1,0) and D=(1,-1).
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Richard J. Mathar, Motzkin Islands: a 3-dimensional Embedding of Motzkin Paths, viXra:2009.0152, 2020.

Crossrefs

Cf. A001006, A333498.

Programs

Maple

P[0]:=1: P[1]:=1-z: for n from 2 to 10 do P[n]:=sort(expand((1-z)*P[n-1]-z^2*P[n-2])) od: for k from 0 to 8 do h[k]:=series(z^(2*k)/P[k]/P[k+1],z=0,20) od: a:=proc(n,k) if k=0 then 1 elif n=0 then 0 else coeff(h[k],z^n) fi end: seq(seq(a(n,k),k=0..floor(n/2)),n=0..15);
# second Maple program:
b:= proc(x, y, h) option remember; `if`(x=0, z^h, add(
      b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..16);  # Alois P. Heinz, Mar 13 2017, revised Mar 28 2020

Mathematica

b[x_, y_, m_] := b[x, y, m] = If[y > x, 0, If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + b[x - 1, y, m] + b[x - 1, y + 1, Max[m, y + 1]]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, May 12 2017, after Alois P. Heinz *)

Formula

The g.f. for column k is z^(2k)/[P_k*P_{k+1}], where the polynomials P_k are defined by P_0=1, P_1=1-z, P_k=(1-z)P_{k-1}-z^2*P_{k-2}.

Sum_{k=1..floor(n/2)} k * T(n,k) = A333498(n). - Alois P. Heinz, Mar 28 2020