A053441 Moments of generalized Motzkin paths.

1, 0, 4, 4, 16, 24, 71, 128, 328, 650, 1552, 3232, 7437, 15904, 35884, 77840, 173792, 379896, 843411, 1851264, 4097552, 9014038, 19918944, 43871360, 96860441, 213472064, 471086932, 1038595100, 2291372912, 5052682904, 11145821407, 24580005376, 54217564504, 119573069218

Offset

2,3

Links

Harvey P. Dale, Table of n, a(n) for n = 2..1000

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Index entries for linear recurrences with constant coefficients, signature (0,4,2,0,0,-1).

Formula

G.f.: x^2*(2*x^3+1)/((1+x)*(1+x-x^2)*(1-2*x-x^3)).

a(n) = 4*a(n-2) + 2*a(n-3) - a(n-6); a(2)=1, a(3)=0, a(4)=4, a(5)=4, a(6)=16, a(7)=24. - Harvey P. Dale, Oct 24 2011

Mathematica

Drop[CoefficientList[Series[x^2(2x^3+1)/((1+x)(1+x-x^2)(1-2x-x^3)), {x, 0, 40}], x], 2] (* or *) LinearRecurrence[{0, 4, 2, 0, 0, -1}, {1, 0, 4, 4, 16, 24}, 40] (* Harvey P. Dale, Oct 24 2011 *)

Prog

(PARI) x='x+O('x^30); Vec(x^2*(2*x^3+1)/((1+x)*(1+x-x^2)*(1-2*x-x^3))) \\ G. C. Greubel, May 26 2018
(Magma) I:=[1, 0, 4, 4, 16, 24]; [n le 6 select I[n] else 4*Self(n-2) + 2*Self(n-3) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 26 2018

Crossrefs

Sequence in context: A223819 A082649 A156232 * A065732 A092959 A330054

Adjacent sequences: A053438 A053439 A053440 * A053442 A053443 A053444

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jan 12 2000

Extensions

More terms from Reiner Martin, Oct 13 2002

Status

approved