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%I #26 Nov 19 2021 10:16:48
%S 1,7,43,249,1395,7653,41381,221399,1175027,6196725,32512401,169863147,
%T 884318973,4589954619,23761814955,122735222505,632698778835,
%U 3255832730565,16728131746145,85826852897675,439793834236745,2251006269442815,11509340056410735,58790764269668805
%N Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).
%C Also a(n) = Sum_{i=0..n} binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i (A056040).
%H Vincenzo Librandi, <a href="/A163869/b163869.txt">Table of n, a(n) for n = 0..300</a>
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html"> Swinging Factorial</a>.
%F From _Vaclav Kotesovec_, Oct 21 2012: (Start)
%F G.f.: -sqrt(x-1)/(5*x-1)^(3/2).
%F Recurrence: n*a(n) = (6*n+1)*a(n-1) - 5*(n-1)*a(n-2).
%F a(n) ~ 4*5^(n-1/2)*sqrt(n)/sqrt(Pi).
%F (End)
%F a(n) = hypergeom([3/2, -n], [1], -4) = hypergeom([3/2, n+1], [1], 4/5)/(5*sqrt(5)). - _Vladimir Reshetnikov_, Apr 25 2016
%F E.g.f.: exp(3*x) * ((1 + 4*x) * BesselI(0,2*x) + 4 * x * BesselI(1,2*x)). - _Ilya Gutkovskiy_, Nov 19 2021
%p a := proc(n) local i; add(binomial(n,i)/Beta(i+1,i+1), i=0..n) end:
%t CoefficientList[Series[-Sqrt[x-1]/(5*x-1)^(3/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 21 2012 *)
%t sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Sum[ Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Jul 26 2013 *)
%t Table[Hypergeometric2F1[3/2, -n, 1, -4], {n, 0, 20}] (* _Vladimir Reshetnikov_, Apr 25 2016 *)
%Y Cf. A163842.
%K nonn
%O 0,2
%A _Peter Luschny_, Aug 06 2009
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