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%I #64 Sep 23 2021 08:57:55
%S 1,-1,2,-3,6,-10,20,-35,70,-126,252,-462,924,-1716,3432,-6435,12870,
%T -24310,48620,-92378,184756,-352716,705432,-1352078,2704156,-5200300,
%U 10400600,-20058300,40116600,-77558760,155117520,-300540195,601080390,-1166803110
%N Inverse binomial transform of A005043.
%C Successive binomial transforms are A005043, A000108, A007317, A064613, A104455. Hankel transform is A000012.
%C Moment sequence of the trace of the square of a random matrix in USp(2)=SU(2). If X=tr(A^2) is a random variable (a distributed with Haar measure) then a(n) = E[X^n]. - _Andrew V. Sutherland_, Feb 29 2008
%C From _Tom Copeland_, Nov 08 2014: (Start)
%C This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Mobius) transformation P(x,t) = x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). The Motzkin sums, or Riordan numbers, A005043 are generated by Mot(x)=C[P(x,1)]. One could, of course, choose the Riordan numbers as the parent sequence.
%C O.g.f.: G(x) = C[P[P(x,1),1]1] = C[P(x,2)] = [1-sqrt(1-4*x/(1+2x)]/2 = x - x^2 + 2 x^3 - ... = Mot[P(x,1)].
%C Ginv(x) = Pinv[Cinv(x),2] = P[Cinv(x),-2] = x(1-x)/[1-2x(1-x)] = (x-x^2)/[1-2(x-x^2)] = x*A146559(x).
%C Cf. A091867 and A210736 for an unsigned version with a leading 1. (End)
%H Vincenzo Librandi, <a href="/A126930/b126930.txt">Table of n, a(n) for n = 0..1000</a>
%H Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.
%F a(n) = (-1)^n*C(n, floor(n/2)) = (-1)^n*A001405(n).
%F a(2*n) = A000984(n), a(2*n+1) = -A001700(n).
%F a(n) = (1/Pi)*Integral_{t=0..Pi}(2cos(2t))^n*2sin^2(t) dt. - _Andrew V. Sutherland_, Feb 29 2008, Mar 09 2008
%F a(n) = (-2)^n + Sum_{k=0..n-1} a(k)*a(n-1-k), a(0)=1. - _Philippe Deléham_, Dec 12 2009
%F G.f.: (1+2*x-sqrt(1-4*x^2))/(2*x*(1+2*x)). - _Philippe Deléham_, Mar 01 2013
%F O.g.f.: (1 + x*c(x^2))/(1 + 2*x), with the o.g.f. c(x) for the Catalan numbers A000108. From the o.g.f. of the Riordan type Catalan triangle A053121. This is the rewritten g.f. given in the previous formula. This is G(-x) with the o.g.f. G(x) of A001405. - _Wolfdieter Lang_, Sep 22 2013
%F D-finite with recurrence (n+1)*a(n) +2*a(n-1) +4*(-n+1)*a(n-2)=0. - _R. J. Mathar_, Dec 04 2013
%F Recurrence (an alternative): (n+1)*a(n) = 8*(n-2)*a(n-3) + 4*(n-2)*a(n-2) + 2*(-n-1)*a(n-1), n>=3. - _Fung Lam_, Mar 22 2014
%F Asymptotics: a(n) ~ (-1)^n*2^(n+1/2)/sqrt(n*Pi). - _Fung Lam_, Mar 22 2014
%F E.g.f.: BesselI(0,2*x) - BesselI(1,2*x). - _Peter Luschny_, Dec 17 2014
%F a(n) = 2^n*hypergeom([3/2,-n], [2], 2). - _Vladimir Reshetnikov_, Nov 02 2015
%F G.f. A(x) satisfies: A(x) = 1/(1 + 2*x) + x*A(x)^2. - _Ilya Gutkovskiy_, Jul 10 2020
%p egf := BesselI(0,2*x) - BesselI(1,2*x):
%p seq(n!*coeff(series(egf,x,34),x,n),n=0..33); # _Peter Luschny_, Dec 17 2014
%t CoefficientList[Series[(1 + 2 x - Sqrt[1 - 4 x^2])/(2 x (1 + 2 x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 23 2013 *)
%t Table[2^n Hypergeometric2F1[3/2, -n, 2, 2], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 02 2015 *)
%o (PARI) x='x+O('x^50); Vec((1+2*x-sqrt(1-4*x^2))/(2*x*(1+2*x))) \\ _Altug Alkan_, Nov 03 2015
%Y Cf. A126120, A126869.
%Y Cf. A000108, A005043, A146559, A091867, A210736.
%K sign
%O 0,3
%A _Philippe Deléham_, Mar 17 2007
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