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%I #27 Jul 01 2020 14:47:13
%S 1,1,2,5,15,51,188,731,2950,12235,51822,223191,974427,4302645,
%T 19181100,86211885,390248055,1777495635,8140539950,37463689775,
%U 173164232965,803539474345,3741930523740,17481709707825,81912506777200,384847173838501,1812610804416698,8556895079642921,40480850291739165,191884148712996795,911225151259732188,4334673398737025619,20653004146207902678,98551406393189773875
%N G.f.: (1/2)*(3 - sqrt((1-5*x)/(1-x))).
%C Same as A007317 if the first 1 is omitted. Has several combinatorial interpretations so deserves its own entry.
%H G. C. Greubel, <a href="/A181768/b181768.txt">Table of n, a(n) for n = 0..1000</a>
%H Ira M. Gessel, Jang Soo Kim, <a href="http://arxiv.org/abs/1003.5301">A note on 2-distant noncrossing partitions and weighted Motzkin paths</a>, arXiv:1003.5301 [math.CO], 2010.
%H Ira M. Gessel, Jang Soo Kim, <a href="http://dx.doi.org/10.1016/j.disc.2010.07.017">A note on 2-distant noncrossing partitions and weighted Motzkin paths</a>, Discrete Math. 310 (2010), no. 23, 3421-3425.
%F a(n) ~ 5^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 29 2013
%F D-finite with recurrence: n*a(n) +2*(-3*n+4)*a(n-1) +5*(n-2)*a(n-2)=0. - _R. J. Mathar_, Aug 06 2013
%F a(n) = JacobiP(n-1,1,-n-1/2,9)/n for n>0. - _Peter Luschny_, Sep 23 2014
%F a(n) = 1 + Sum_{k=1..n-1} a(k) * a(n-k). - _Ilya Gutkovskiy_, Jul 01 2020
%p A181768 := n -> `if`(n=0, 1, JacobiP(n-1,1,-n-1/2,9)/n):
%p seq(round(evalf(A181768(n),99)), n=0..33); # _Peter Luschny_, Sep 23 2014
%t CoefficientList[Series[3/2-Sqrt[(1-5x)/(1-x)]/2,{x,0,40}],x] (* _Harvey P. Dale_, Jul 28 2013 *)
%o (PARI) x='x + O('x^50); Vec((1/2)*(3 - sqrt((1-5*x)/(1-x)))) \\ _G. C. Greubel_, Feb 12 2017
%Y Cf. A007317.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Nov 12 2010
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