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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)). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

LINKS

Table of n, a(n) for n=0..59.

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.

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Dec 04 2017, after Peter Luschny *)

CROSSREFS

Cf. A001006, A004148, A001263.

Sequence in context: A089886 A071511 A119922 * A098979 A071928 A165207

Adjacent sequences:  A097857 A097858 A097859 * A097861 A097862 A097863

KEYWORD

nonn,tabf,changed

AUTHOR

Emeric Deutsch, Sep 01 2004

STATUS

approved

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Last modified February 21 19:50 EST 2018. Contains 299423 sequences. (Running on oeis4.)