A097885 -id:A097885 - cp's OEIS frontend

A097860 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n with k peaks (n>=0, 0<=k<=floor(n/2)).

1, 1, 1, 1, 2, 2, 4, 4, 1, 8, 10, 3, 17, 24, 9, 1, 37, 58, 28, 4, 82, 143, 81, 16, 1, 185, 354, 231, 60, 5, 423, 881, 653, 205, 25, 1, 978, 2204, 1824, 676, 110, 6, 2283, 5534, 5058, 2164, 435, 36, 1, 5373, 13940, 13946, 6756, 1631, 182, 7, 12735, 35213, 38262, 20710

Offset: 0

Emeric Deutsch, Sep 01 2004

Row sums are the Motzkin numbers (A001006). Column 0 gives A004148.

This triangle is the Motzkin path equivalent to the Narayana numbers (A001263). - Dan Drake, Feb 17 2011

			Triangle starts:
   1;
   1;
   1,  1;
   2,  2;
   4,  4, 1;
   8, 10, 3;
  17, 24, 9, 1;
  ...
Row n has 1+floor(n/2) terms.
T(4,1)=4 because (UD)HH, H(UD)H, HH(UD) and U(UD)D are the only Motzkin paths of length 4 with 1 peak (here U=(1,1), H=(1,0) and D=(1,-1)); peaks are shown between parentheses.

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.
Dan Drake and Ryan Gantner, Generating functions for plateaus in Motzkin paths, J. of the Chungcheong Math. Soc., Vol 25, No. 3, p. 475, August 2012.
Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.

Cf. A001006, A004148, A001263, A097885.

eq:=G=1+z*G+z^2*G*(G-1+t):sol:=solve(eq,G): G:=sol[2]: Gser:=simplify(series(G,z=0,16)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od: seq(seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)),n=0..13);
# Alternatively
A097860_row := proc(n) local w,f,p,i;
w := 1-x+x^2-x^2*t; f := (w-sqrt(w^2-4*x^2))/(2*x^2);
p := simplify(coeff(series(f,x,3+2*n),x,n));
seq(coeff(p,t,i), i=0..iquo(n,2)) end:
seq(print(A097860_row(n)), n=0..7); # Peter Luschny, Nov 14 2014
# third Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 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

gf = With[{w = 1 - x + x^2 - x^2*t}, (w - Sqrt[w^2 - 4*x^2])/(2*x^2)];
cx[n_] := cx[n] = SeriesCoefficient[gf, {x, 0, n}];
T[n_, k_] := SeriesCoefficient[cx[n], {t, 0, k}];
Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Dec 04 2017, after Peter Luschny *)

G.f. G = G(t, z) satisfies G = 1+z*G+z^2*G*(G-1+t).

G.f. has explicit form G(x,t) = (w-sqrt(w^2-4*x^2))/(2*x^2) with w = 1-x+x^2-x^2*t. (Drake and Ganter, Th. 6) - Peter Luschny, Nov 14 2014

cp's OEIS Frontend

A097874 Duplicate of A097885.

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A097860 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n with k peaks (n>=0, 0<=k<=floor(n/2)).

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