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%I #38 Jul 06 2023 01:56:48
%S 1,2,6,18,56,176,558,1778,5686,18230,58558,188366,606588,1955044,
%T 6305418,20347342,65689088,212146400,685342218,2214556478,7157409064,
%U 23136645472,74801223162,241863933094,782131232390,2529458676326
%N Expansion of 1/(sqrt(1-2x-3x^2)-x).
%C Row sums of A111960.
%C A transform of the Fibonacci numbers. - _Paul Barry_, Sep 23 2005
%C Apparently the Motzkin transform of (0 followed by A128588). - _R. J. Mathar_, Dec 11 2008
%C Inverse binomial transform of A026671. - _Philippe Deléham_, Feb 11 2009
%C Hankel transform is 2^n. - _Paul Barry_, Mar 02 2010
%H Vincenzo Librandi, <a href="/A111961/b111961.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H Paul Barry, <a href="https://arxiv.org/abs/2307.00098">Moment sequences, transformations, and Spidernet graphs</a>, arXiv:2307.00098 [math.CO], 2023.
%H Taras Goy and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Shattuck/sh36.html">Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries</a>, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C((j-1)/2, (j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
%F a(n) = Sum_{k=0..n} F(k+1)*Sum_{i=0..floor((n-k)/2)} C(n, i)*C(n-i, i+k)/(i+k+1). - _Paul Barry_, Sep 23 2005
%F G.f.: M(x)^2/(2*M(x)-M(x)^2), where M(x) is the g.f. of the Motzkin numbers A001006. - _Paul Barry_, Feb 03 2006
%F G.f.: 1/(1-2x/(1-x/(1-x^2/(1-x/(1-x/91-x^2/(1-x/(1-x/(1-x^2/(1-... (continued fraction). - _Paul Barry_, Mar 02 2010
%F D-finite with recurrence: n*a(n) + (-4*n+3)*a(n-1) + 3*(-n+1)*a(n-2) + 2*(7*n-15)*a(n-3) + 12*(n-3)*a(n-4) = 0. - _R. J. Mathar_, Nov 15 2012
%F a(n) ~ (1+sqrt(5))^n / sqrt(5). - _Vaclav Kotesovec_, Feb 08 2014
%t CoefficientList[Series[1/(Sqrt[1-2*x-3*x^2]-x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)
%Y Cf. A001006, A026671, A111960, A128588.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Aug 23 2005
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