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A242233 2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.

%I A242233 #16 Nov 05 2024 19:30:52
%S A242233 1,1,3,11,41,137,347,611,5777,98321,677363,-4192197,-134908871,
%T A242233 -617972327,22749265099,449951818387,-632325203423,-163681108703199,
%U A242233 -2324079456844573,33233931805782635,1734259111955765577,14135975420529458857,-777499293367428199109
%N A242233 2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.
%F A242233 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).
%F A242233 For n > 0, a(n) = Sum_{k=1..n} a(n-k)*binomial(n,k)*(2*k)!*(3*k/(2*n)-1)*2^k/(k!*(k+1)!). - _Tani Akinari_, Nov 05 2024
%p A242233 f := sqrt(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)));
%p A242233 seq(2^n*n!*coeff(series(f,x,n+1),x,n),n=0..22);
%p A242233 # Second program with function g from A241885:
%p A242233 catalan := n -> binomial(2*n,n)/(n+1);
%p A242233 a := n -> 2^n*g(catalan, n); seq(a(n), n=0..22);
%t A242233 g[n_] := g[n] = (CatalanNumber[n] - Sum[Binomial[n, m] g[m] g[n - m], {m, 1, n - 1}])/2;
%t A242233 a[0] = 1; a[n_] := 2^n g[n];
%t A242233 Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Aug 02 2019, from 2nd Maple program *)
%o A242233 (Maxima) a[n]:=if n=0 then 1 else sum(a[n-k]*binomial(n, k)*(2*k)!*(3*k/(2*n)-1)*2^k/(k!*(k+1)!), k, 1, n); makelist(a[n],n,0,50); /* _Tani Akinari_, Nov 05 2024 */
%Y A242233 Cf. A000108, A126156, A241885.
%K A242233 sign
%O A242233 0,3
%A A242233 _Peter Luschny_, May 08 2014