login
Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).
2

%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