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A126087 Expansion of c(2x^2)/(1-xc(2x^2)), where c(x) = (1-sqrt(1-4x))/(2x) is the g.f. of the Catalan numbers (A000108). 8

%I

%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(2x^2)/(1-xc(2x^2)), where c(x) = (1-sqrt(1-4x))/(2x) is the g.f. of the Catalan numbers (A000108).

%C Series reversion of x(1+x)/(1+2x+3x^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 A. Bostan, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.410.1160&amp;rep=rep1&amp;type=pdf">Computer Algebra for Lattice Path Combinatorics</a>, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

%F G.f.: [1-sqrt(1-8x^2)]/[x(4x-1+sqrt(1-8x^2))]. - _Emeric Deutsch_, Mar 04 2007

%F a(n) = Sum_{k, 0<=k<=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(2n) = A089022(n). - _Philippe Deléham_, Nov 02 2011

%F Conjecture: (n+1)*a(n) -3*(n+1)*a(n-1) +8*(2-n)*a(n-2) +24*(n-2)*a(n-3) =0. - _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 *)

%Y Cf. A000108.

%K nonn

%O 0,3

%A _Philippe Deléham_, Mar 03 2007

%E More terms from _Emeric Deutsch_, Mar 04 2007

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Last modified February 17 17:52 EST 2019. Contains 320222 sequences. (Running on oeis4.)