A168049 Expansion of (3 -x -sqrt(1-2*x-3*x^2))/2.

1, 0, 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209

Offset: 0

Author

Paul Barry, Nov 17 2009

Keywords

• easy
• nonn

Comments

A variant of the Motzkin numbers A001006. Hankel transform is A168050.

Essentially the same as A086246. - R. J. Mathar, Dec 20 2011

Alternatively, this sequence corresponds to the number of positive walks with n steps {-1,0,1} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - David Nguyen, Dec 01 2016

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 51*x^8 + ... - _Michael Somos_, Sep 26 2018
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Crossrefs

Cf. A001006, A086246, A168050, A168051.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3 -x - Sqrt(1-2*x-3*x^2))/2)); // G. C. Greubel, Sep 25 2018

Mathematica

CoefficientList[Series[(3-x-Sqrt[1-2*x-3*x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

PARI

Vec((3-x-sqrt(1-2*x-3*x^2))/2) \\ Charles R Greathouse IV, Dec 01 2016

Formula

D-finite with recurrence: n*a(n) +(3-2n)*a(n-1) +3(3-n)*a(n-2)=0. - R. J. Mathar, Dec 20 2011

0 = a(n)*(+9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1)*(-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 31 2014

a(n) ~ 3^(n+1/2) / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

G.f.: 1 + x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Sep 23 2017