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A168049 Expansion of (3-x-sqrt(1-2x-3x^2))/2. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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. - D. Nguyen, Dec 01 2016

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.

FORMULA

Conjecture: 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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A168051.

Sequence in context: A094288 A168051 A166587 * A001006 A086246 A247100

Adjacent sequences:  A168046 A168047 A168048 * A168050 A168051 A168052

KEYWORD

easy,nonn,changed

AUTHOR

Paul Barry, Nov 17 2009

STATUS

approved

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Last modified December 10 12:03 EST 2016. Contains 279002 sequences.