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%I #32 Jul 22 2022 17:45:30
%S 1,2,5,14,40,116,339,996,2937,8684,25729,76352,226868,674806,2008907,
%T 5984886,17841024,53212500,158784033,473995320,1415449578,4228149450,
%U 12633596331,37758241434,112873961079,337492122822,1009283640669,3018807519506,9030752740042
%N Number of different compact source directed animals with 1 point on the bottom line.
%H Harvey P. Dale, <a href="/A036908/b036908.txt">Table of n, a(n) for n = 1..1000</a>
%H Nickolas Hein and Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%F G.f.: x^2*(1-3*x+sqrt(1-2*x-3*x^2))/((1-3*x)*(3*x-1+sqrt(1-2*x-3*x^2))). - amended by _Georg Fischer_, Apr 09 2020
%F D-finite with recurrence: (-n+1)*a(n) +(5*n-8)*a(n-1) +3*(-n+1)*a(n-2) +9*(-n+4)*a(n-3)=0. - _R. J. Mathar_, Jan 28 2020
%F a(n) ~ 3^(n-2) * (1 + sqrt(3/(Pi*n))). - _Vaclav Kotesovec_, Apr 09 2020
%t Rest[CoefficientList[Series[x^2*(1-3*x+Sqrt[1-2*x-3*x^2])/((1-3*x)*(3x-1+Sqrt[1- 2*x-3*x^2])), {x,0,30}], x]] (* _Harvey P. Dale_, Jun 03 2012; _Georg Fischer_, Apr 09 2020 *)
%o (PARI) seq(n)={my(p=sqrt(1-2*x-3*x^2 + O(x*x^n))); Vec(x^2*(1-3*x+p)/((1-3*x)*(3*x-1+p)))} \\ _Andrew Howroyd_, Apr 09 2020
%K nonn
%O 1,2
%A _Louis Shapiro_
%E More terms from _Harvey P. Dale_, Jun 03 2012
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