A167022 Expansion of sqrt(1 - 2*x - 3*x^2) in powers of x.

1, -1, -2, -2, -4, -8, -18, -42, -102, -254, -646, -1670, -4376, -11596, -31022, -83670, -227268, -621144, -1706934, -4713558, -13072764, -36398568, -101704038, -285095118, -801526446, -2259520830, -6385455594, -18086805002, -51339636952, -146015545604

Offset

0,3

Comments

Sequence is to Motzkin numbers as A002420 is to Catalan numbers.

Links

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

Formula

D-finite with recurrence: n*a(n) = (2*n - 3)*a(n-1) + (3*n - 9)*a(n-2) for n>1.

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)) for all n in Z. - Michael Somos, Mar 23 2012

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

Convolution inverse of A002426. A007971(n) = -a(n) unless n=0. A126068(n) = -a(n) unless n=0 or n=1. A001006(n) = -a(n+2)/2 unless n=0 or n=1.

G.f.: A(x)=sqrt(1-2*a*x+((a)^2-4*b)*(x^2)) =1-a*x-2*b*x^2/G(0) ; G(k) = 1 - a*x - b*x^2/G(k+1). - Sergei N. Gladkovskii, Dec 05 2011

a=1;b=1;A(x)=(1-2*x-3*x^2)^(1/2)=1-x-2*x^2/G(0) ; G(k) = 1 - x - x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011

G.f.: sqrt(1-2*x-3*(x^2))=1 - x/G(0) = (3*x+2)*G(0) - 1 ; G(k) = 1 - 2*x/(1 + x/(1 + x/(1 - 2*x/(1 - x/(2 - x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011

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

a(n) = 2*A168051(n), n>1. - R. J. Mathar, Jan 23 2020

Example

G.f. = 1 - x - 2*x^2 - 2*x^3 - 4*x^4 - 8*x^5 - 18*x^6 - 42*x^7 - 102*x^8 + ...

Mathematica

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

Prog

(PARI) {a(n) = polcoeff( sqrt(1 - 2*x - 3*x^2 + x * O(x^n)), n)}

Crossrefs

Cf. A001006, A002426, A007971, A126068.

Sequence in context: A168058 A007971 A126068 * A168055 A005702 A095335

Adjacent sequences: A167019 A167020 A167021 * A167023 A167024 A167025

Keyword

sign

Author

Michael Somos, Oct 27 2009

Status

approved