A244884 Expansion of (-2 +x^2 +x -x*sqrt(1-2*x-3*x^2))/(-1 +x -sqrt(1-2*x-3*x^2)).

1, 1, 1, 2, 5, 12, 30, 76, 196, 512, 1353, 3610, 9713, 26324, 71799, 196938, 542895, 1503312, 4179603, 11662902, 32652735, 91695540, 258215664, 728997192, 2062967382, 5850674704, 16626415975, 47337954326, 135015505407, 385719506620, 1103642686382

Offset: 0

Author

N. J. A. Sloane, Jul 09 2014

Keywords

• nonn

Comments

For n > 1, a(n) is the number of Motzkin n-paths that start with an up step. - Gennady Eremin, Sep 18 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• J.-L. Baril and A. Petrossian, Equivalence classes of Dyck paths modulo some statistics, 2014.
• Gennady Eremin, Generating function for Naturalized Series: The case of Ordered Motzkin Words, arXiv:2002.08067 [math.CO], 2020.

Crossrefs

Apart from initial terms, same as A002026 and A105695.

Cf. A001006.

Programs

Mathematica

CoefficientList[Series[(-2 + x^2 + x - x Sqrt[1 - 2 x - 3 x^2])/(-1 + x - Sqrt[1 - 2 x - 3 x^2]), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 10 2014 *)

PARI

my(x='x + O('x^50)); Vec((-2 +x^2 +x -x*sqrt(1-2*x-3*x^2))/(-1 +x -sqrt(1-2*x-3*x^2))) \\ G. C. Greubel, Feb 14 2017

Formula

a(n) ~ 3^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 10 2014

Conjecture D-finite with recurrence: (n+2)*a(n) +(-3*n-1)*a(n-1) -n*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Jan 24 2020

G.f.: x + (1-x)*M(x), where M(x) is the g.f. of A001006. - Gennady Eremin, Feb 14 2021