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%I #29 Apr 21 2020 19:06:39
%S 1,2,7,26,106,452,1999,9074,42046,198044,945430,4564100,22243060,
%T 109285256,540738943,2692103714,13475973238,67784600108,342439638418,
%U 1736727343436,8839203054604,45132514680248,231121351433158
%N Expansion of (1 - 2*x - sqrt(1 - 4*x - 8*x^2))/(6*x^2).
%C Series reversion of x/(1+2x+3x^2). Binomial transform is A107264. Counts colored Motzkin paths. Second binomial transform of 1,0,3,0,18,0,... or 3^n*binomial(n) (A005159) with interpolated zeros.
%C Hankel transform is 3^binomial(n+1,2). - _Paul Barry_, Oct 01 2009
%H G. C. Greubel, <a href="/A122871/b122871.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%F E.g.f.: exp(2*x)*Bessel_I(1, sqrt(3)*2*x)/(sqrt(3)x).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*binomial(k)3^k*2^(n-2k).
%F G.f.: 1/(1-2x-3x^2/(1-2x-3x^2/(1-2x-3x^2/(1-2x-3x^2/(1-.... (continued fraction). - _Paul Barry_, Oct 01 2009
%F D-finite with recurrence: (n+2)*a(n) - 2*(2n+1)*a(n-1) + 8*(1-n)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011
%F a(n) ~ 2*sqrt(9+5*sqrt(3))*(2+2*sqrt(3))^n/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012
%t CoefficientList[Series[(1-2*x-Sqrt[1-4*x-8*x^2])/(6*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o (Sage)
%o def A122871_list(n): # n>=1
%o T = [0]*(n+1); R = [1]
%o for m in (1..n-1):
%o a,b,c = 1,0,0
%o for k in range(m,-1,-1):
%o r = a + 2*b + 3*c
%o if k < m : T[k+2] = u;
%o a,b,c = T[k-1],a,b
%o u = r
%o T[1] = u; R.append(u)
%o return R
%o A122871_list(23) # _Peter Luschny_, Nov 01 2012
%o (PARI) x='x+O('x^50); Vec((1 - 2*x - sqrt(1 - 4*x - 8*x^2))/(6*x^2)) \\ _G. C. Greubel_, Mar 19 2017
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 16 2006