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

%I A167022 #37 Jan 30 2020 21:29:16
%S A167022 1,-1,-2,-2,-4,-8,-18,-42,-102,-254,-646,-1670,-4376,-11596,-31022,
%T A167022 -83670,-227268,-621144,-1706934,-4713558,-13072764,-36398568,
%U A167022 -101704038,-285095118,-801526446,-2259520830,-6385455594,-18086805002,-51339636952,-146015545604
%N A167022 Expansion of sqrt(1 - 2*x - 3*x^2) in powers of x.
%C A167022 Sequence is to Motzkin numbers as A002420 is to Catalan numbers.
%H A167022 G. C. Greubel, <a href="/A167022/b167022.txt">Table of n, a(n) for n = 0..1000</a>
%F A167022 D-finite with recurrence: n*a(n) = (2*n - 3)*a(n-1) + (3*n - 9)*a(n-2) for n>1.
%F A167022 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
%F A167022 G.f.: sqrt(1 - 2*x - 3*x^2).
%F A167022 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.
%F A167022 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
%F A167022 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
%F A167022 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
%F A167022 a(n) ~ -3^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 05 2018
%F A167022 a(n) = 2*A168051(n), n>1. - _R. J. Mathar_, Jan 23 2020
%e A167022 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 + ...
%t A167022 a[ n_] := SeriesCoefficient[ Sqrt[1 - 2 x - 3 x^2], {x, 0, n}] (* _Michael Somos_, Jan 25 2014 *)
%o A167022 (PARI) {a(n) = polcoeff( sqrt(1 - 2*x - 3*x^2 + x * O(x^n)), n)}
%Y A167022 Cf. A001006, A002426, A007971, A126068.
%K A167022 sign
%O A167022 0,3
%A A167022 _Michael Somos_, Oct 27 2009