A053442 Moments of generalized Motzkin paths.

1, 0, 2, 1, 6, 6, 21, 30, 82, 141, 342, 650, 1485, 2982, 6612, 13693, 29922, 63072, 136905, 291618, 631302, 1353441, 2928054, 6303798, 13642117, 29454702, 63791456, 138020533, 299191968, 648376932, 1406836717, 3052671816, 6629649798, 14400972413, 31301837952

Offset: 0

Author

N. J. A. Sloane, Jan 12 2000

Keywords

• nonn
• easy
• nice

Comments

From Seiichi Manyama, Apr 30 2025: (Start)

Number of lattice paths from (0,0) to (n,n) using steps (2,0),(0,2),(3,3).

Diagonal of the rational function 1 / (1 - x^2 - y^2 - x^3*y^3).

Diagonal of the rational function 1 / ((1-x^2*y)*(1-x*y^2) - y). (End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2917
• R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Crossrefs

Cf. A002426.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1 - 4 x^2 - 2 x^3 + x^6], {x, 0, 34}], x], (* Michael De Vlieger, Dec 25 2021 *)

PARI

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^2-2*x^3+x^6)) \\ Seiichi Manyama, Apr 30 2025

Formula

G.f.: 1 / sqrt(1-4*z^2-2*z^3+z^6). - Sean A. Irvine, Dec 25 2021

Extensions

More terms from Reiner Martin, Oct 13 2002

Typos in terms corrected by Sean A. Irvine, Dec 25 2021

Offset changed to 0 by Seiichi Manyama, Apr 30 2025