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A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

%I A163869 #38 Aug 22 2025 17:19:50
%S A163869 1,7,43,249,1395,7653,41381,221399,1175027,6196725,32512401,169863147,
%T A163869 884318973,4589954619,23761814955,122735222505,632698778835,
%U A163869 3255832730565,16728131746145,85826852897675,439793834236745,2251006269442815,11509340056410735,58790764269668805
%N A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).
%C A163869 Also a(n) = Sum_{i=0..n} binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i (A056040).
%H A163869 Vincenzo Librandi, <a href="/A163869/b163869.txt">Table of n, a(n) for n = 0..300</a>
%H A163869 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163869 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163869 From _Vaclav Kotesovec_, Oct 21 2012: (Start)
%F A163869 G.f.: -sqrt(x-1)/(5*x-1)^(3/2).
%F A163869 Recurrence: n*a(n) = (6*n+1)*a(n-1) - 5*(n-1)*a(n-2).
%F A163869 a(n) ~ 4*5^(n-1/2)*sqrt(n)/sqrt(Pi).
%F A163869 (End)
%F A163869 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 A163869 E.g.f.: exp(3*x) * ((1 + 4*x) * BesselI(0,2*x) + 4 * x * BesselI(1,2*x)). - _Ilya Gutkovskiy_, Nov 19 2021
%F A163869 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A163869 a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A163869 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
%F A163869 a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)
%p A163869 a := proc(n) local i; add(binomial(n,i)/Beta(i+1,i+1), i=0..n) end:
%t A163869 CoefficientList[Series[-Sqrt[x-1]/(5*x-1)^(3/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 21 2012 *)
%t A163869 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 A163869 Table[Hypergeometric2F1[3/2, -n, 1, -4], {n, 0, 20}] (* _Vladimir Reshetnikov_, Apr 25 2016 *)
%Y A163869 Cf. A163842, A163872, A387208.
%K A163869 nonn,changed
%O A163869 0,2
%A A163869 _Peter Luschny_, Aug 06 2009