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%I #10 Aug 02 2019 12:42:59
%S 1,1,3,11,41,137,347,611,5777,98321,677363,-4192197,-134908871,
%T -617972327,22749265099,449951818387,-632325203423,-163681108703199,
%U -2324079456844573,33233931805782635,1734259111955765577,14135975420529458857,-777499293367428199109
%N 2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.
%F a(n) = 2^n*n!*[x^n](sqrt(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)))), where [x^n](f(x)) the coefficient of x^n in f(x).
%p f := sqrt(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)));
%p seq(2^n*n!*coeff(series(f,x,n+1),x,n),n=0..22);
%p # Second program with function g from A241885:
%p catalan := n -> binomial(2*n,n)/(n+1);
%p a := n -> 2^n*g(catalan, n); seq(a(n), n=0..22);
%t g[n_] := g[n] = (CatalanNumber[n] - Sum[Binomial[n, m] g[m] g[n - m], {m, 1, n - 1}])/2;
%t a[0] = 1; a[n_] := 2^n g[n];
%t Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Aug 02 2019, from 2nd Maple program *)
%Y Cf. A126156, A241885.
%K sign
%O 0,3
%A _Peter Luschny_, May 08 2014
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