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URL: https://oeis.org/A053442

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A053442
Moments of generalized Motzkin paths.
6
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
0,3
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
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
FORMULA
G.f.: 1 / sqrt(1-4*z^2-2*z^3+z^6). - Sean A. Irvine, Dec 25 2021
MATHEMATICA
CoefficientList[Series[1/Sqrt[1 - 4 x^2 - 2 x^3 + x^6], {x, 0, 34}], x], (* Michael De Vlieger, Dec 25 2021 *)
PROG
(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
CROSSREFS
Cf. A002426.
Sequence in context: A062820 A113336 A113979 * A019082 A345323 A052636
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jan 12 2000
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
STATUS
approved