%I #41 Aug 06 2024 04:47:25
%S 1,1,3,5,15,29,87,181,543,1181,3543,7941,23823,54573,163719,381333,
%T 1143999,2699837,8099511,19319845,57959535,139480397,418441191,
%U 1014536117,3043608351,7426790749,22280372247,54669443141,164008329423
%N Expansion of c(2*x^2)/(1-x*c(2*x^2)), where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).
%C Series reversion of x*(1+x)/(1+2*x+3*x^2) [offset 0]. - _Paul Barry_, Mar 13 2007
%C Hankel transform is 2^C(n+1,2). - _Philippe Deléham_, Mar 16 2007
%H Vincenzo Librandi, <a href="/A126087/b126087.txt">Table of n, a(n) for n = 0..200</a>
%H Alin Bostan, <a href="https://citeseerx.ist.psu.edu/pdf/749aef4c6f3668e652b5074e5268346ccecc88c9">Computer Algebra for Lattice Path Combinatorics</a>, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
%H Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, <a href="https://arxiv.org/abs/2001.00393">Stieltjes moment sequences for pattern-avoiding permutations</a>, arXiv:2001.00393 [math.CO], 2020.
%F G.f.: (1-sqrt(1-8*x^2))/(x*(4*x-1+sqrt(1-8*x^2))). - _Emeric Deutsch_, Mar 04 2007
%F a(n) = Sum_{k=0..n} 2^(n-k)*A120730(n,k). - _Philippe Deléham_, Oct 16 2008
%F a(n) = Sum_(k=1..n} (1+(-1)^(n-k))*k*2^((n-k)/2-1)*C(n,(n+k)/2)/n, n>0. - _Vladimir Kruchinin_, Feb 18 2011
%F a(2*n) = A089022(n). - _Philippe Deléham_, Nov 02 2011
%F D-finite with recurrence: (n+1)*a(n) = 3*(n+1)*a(n-1) - 8*(2-n)*a(n-2) - 24*(n-2)*a(n-3). - _R. J. Mathar_, Nov 14 2011
%F a(n) ~ 2^(3*(n+1)/2) * (3+2*sqrt(2) + (3-2*sqrt(2))*(-1)^n) / (n^(3/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Feb 13 2014
%p c:=x->(1-sqrt(1-4*x))/2/x: G:=c(2*x^2)/(1-x*c(2*x^2)): Gser:=series(G,x=0,35): seq(coeff(Gser,x,n),n=0..32); # _Emeric Deutsch_, Mar 04 2007
%t CoefficientList[Series[(1-Sqrt[1-8*x^2])/(x*(4*x-1+Sqrt[1-8*x^2])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-8*x^2))/(x*(4*x-1+Sqrt(1-8*x^2))) )); // _G. C. Greubel_, Nov 07 2022
%o (SageMath)
%o def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
%o def A126087(n): return sum(2^(n-k)*A120730(n,k) for k in range(n+1))
%o [A126087(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2022
%Y Cf. A000108, A089022, A120730.
%K nonn
%O 0,3
%A _Philippe Deléham_, Mar 03 2007
%E More terms from _Emeric Deutsch_, Mar 04 2007