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%I #42 Sep 08 2022 08:45:15
%S 1,1,0,1,-1,3,-6,15,-36,91,-232,603,-1585,4213,-11298,30537,-83097,
%T 227475,-625992,1730787,-4805595,13393689,-37458330,105089229,
%U -295673994,834086421,-2358641376,6684761125,-18985057351,54022715451,-154000562758,439742222071,-1257643249140
%N Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)).
%C Binomial transform is A072100.
%C Signed Motzkin numbers with an additional leading 1.
%C Inverse binomial transform of A001405 gives this without the initial 1. So does the binomial transform of (-1)^n*A000108(n) = [1,-1,2,-5,14,-42,...]. - _Philippe Deléham_, Mar 20 2007
%H G. C. Greubel, <a href="/A099323/b099323.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC'02 Melbourne, 2002.
%F a(n) = 0^n + Sum_{k=0..n-1} binomial(n-1,k)*(-1)^k*C(k), where C(k) is the k-th Catalan number.
%F G.f.: 1 + x/(1-sqrt(x))/G(0), where G(k)= 1 + sqrt(x)/(1 - sqrt(x)/(1 + x/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 28 2013
%F D-finite with recurrence: n*a(n) + 2*(n-2)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - _R. J. Mathar_, Oct 10 2014
%F a(n) ~ -(-1)^n * 3^(n + 1/2) / (8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 31 2017
%p with(PolynomialTools): CoefficientList(convert(taylor((sqrt(1 + 3*x) + sqrt(1 - x))/2/sqrt(1 - x), x = 0, 33), polynom), x); # _Taras Goy_, Aug 07 2017
%t CoefficientList[Series[(Sqrt[1+3x]+Sqrt[1-x])/(2Sqrt[1-x]),{x,0,40}],x] (* _Harvey P. Dale_, Feb 06 2015 *)
%o (Magma)
%o A099323:= func< n | (&+[(-1)^k*Binomial(n-1, k)*Catalan(k): k in [0..n]]) >;
%o [A099323(n): n in [0..40]]; // _G. C. Greubel_, Nov 25 2021
%o (Sage) [sum((-1)^k*binomial(n-1, k)*catalan_number(k) for k in (0..n)) for n in (0..40)] # _G. C. Greubel_, Nov 25 2021
%Y Cf. A000108, A001405, A005043, A072100.
%K easy,sign
%O 0,6
%A _Paul Barry_, Oct 12 2004
%E Edited by _N. J. A. Sloane_, Oct 05 2009
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