A168051 Expansion of (1+x+sqrt(1-2x-3x^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

Offset

0,5

Comments

A signed variant of the Motzkin numbers A001006. Hankel transform is A168052.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

D-finite with recurrence: n*a(n) -(2n-3)*a(n-1) -3*(n-3)*a(n-2)=0 if n>2. - _R. J. Mathar_, Dec 20 2011 [Edited by _Michael Somos_, Jan 25 2014]

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 25 2014

G.f.: 1 + x - (x + x^2) / (1 + x - (x + x^2) / (1 + x - ...)). - _Michael Somos_, Mar 27 2014

Convolution inverse of A005043. - _Michael Somos_, Mar 27 2014

a(n) ~ -3^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 05 2018

From _Gennady Eremin_, Feb 25 2021: (Start)

For n > 1, a(n) = A167022(n) / 2.

G.f.: (1 + x + A(x)) / 2, where A(x) is the g.f. of A167022. (End)

Example

G.f. = 1 - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 21*x^7 - 51*x^8 - 127*x^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1 + x + Sqrt[1 - 2 x - 3 x^2]) / 2, {x, 0, n}] (* _Michael Somos_, Jan 25 2014 *)

Prog

(PARI) {a(n) = polcoeff( (1 + x + sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2, n)}; /* _Michael Somos_, Jan 25 2014 */

Crossrefs

Cf. A168049, A166587, A167022.

Cf. A001006, A005043.

Sequence in context: A094286 A094287 A094288 * A166587 A292440 A168049

Adjacent sequences: A168048 A168049 A168050 * A168052 A168053 A168054

Keyword

easy,sign

Author

_Paul Barry_, Nov 17 2009

Status

approved