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A071717 Expansion of (1 + x^2*C)*C^2, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.

%I A071717 #13 May 30 2020 19:12:35
%S A071717 1,2,6,17,51,160,519,1727,5863,20228,70720,250002,892126,3209328,
%T A071717 11626385,42378075,155307615,571925820,2115257100,7853744910,
%U A071717 29263124250,109384710240,410075910270,1541481197334,5808790935126
%N A071717 Expansion of (1 + x^2*C)*C^2, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.
%H A071717 Michael De Vlieger, <a href="/A071717/b071717.txt">Table of n, a(n) for n = 0..1000</a>
%H A071717 Stoyan Dimitrov, <a href="https://arxiv.org/abs/2002.12322">On permutation patterns with constrained gap sizes</a>, arXiv:2002.12322 [math.CO], 2020.
%F A071717 Conjecture: (n+2)*a(n) +(-3*n-2)*a(n-1) +(-5*n+8)*a(n-2) +2*(2*n-7)*a(n-3)=0. - _R. J. Mathar_, Aug 25 2013
%F A071717 G.f.: ( (1 -x -3*x^2) - (1 +x -x^2)*sqrt(1-4*x) )/(2*x^2). - _G. C. Greubel_, May 30 2020
%p A071717 seq(coeff(series( ((1-x-3*x^2) -(1+x-x^2)*sqrt(1-4*x))/(2*x^2) , x, n+1), x, n), n = 0..30); # _G. C. Greubel_, May 30 2020
%t A071717 With[{$MaxExtraPrecision = 1000}, CoefficientList[Series[(1 + x^2*#)*#^2 &[(1 - (1 - 4 x)^(1/2))/(2 x)], {x, 0, 24}], x]] (* _Michael De Vlieger_, May 30 2020 *)
%o A071717 (Sage)
%o A071717 def A071717_list(prec):
%o A071717     P.<x> = PowerSeriesRing(ZZ, prec)
%o A071717     return P( ((1-x-3*x^2) -(1+x-x^2)*sqrt(1-4*x))/(2*x^2) ).list()
%o A071717 A071717_list(30) # _G. C. Greubel_, May 30 2020
%Y A071717 Cf. A000108, A071716, A071718, A071719.
%K A071717 nonn
%O A071717 0,2
%A A071717 _N. J. A. Sloane_, Jun 06 2002