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A163865 The binomial transform of the swinging factorial (A056040).

%I A163865 #20 May 08 2020 17:45:11
%S A163865 1,2,5,16,47,146,447,1380,4251,13102,40343,124136,381625,1172198,
%T A163865 3597401,11031012,33798339,103477590,316581567,967900224,2957316429,
%U A163865 9030317478,27558851565,84059345244,256265811333,780885245826,2378410969977,7241027262280
%N A163865 The binomial transform of the swinging factorial (A056040).
%C A163865 a(n) = Sum_{k=0..n} binomial(n,k) * k$, where k$ denotes the swinging factorial of k (A056040). The swinging analog to the number of arrangements, the binomial transform of the factorial (A000522).
%H A163865 G. C. Greubel, <a href="/A163865/b163865.txt">Table of n, a(n) for n = 0..1000</a>
%H A163865 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163865 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163865 E.g.f.: exp(x)*BesselI(0,2*x)*(1+x). - _Peter Luschny_, Aug 26 2012
%F A163865 O.g.f.: (1-x-4*x^2)/((1+x)*(1-3*x))^(3/2). - _Peter Luschny_, Oct 31 2013
%F A163865 a(n) ~ 3^(n - 1/2) * sqrt(n) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Nov 27 2017
%p A163865 a := proc(n) local k: add(binomial(n,k)*(k!/iquo(k, 2)!^2),k=0..n) end:
%p A163865 seq(coeff(series((1-z-4*z^2)/((1+z)*(1-3*z))^(3/2),z,28),z,n),n=0..27); # _Peter Luschny_, Oct 31 2013
%t A163865 sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Sum[Binomial[n, k]*sf[k], {k, 0, n}]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jul 26 2013 *)
%t A163865 sf[n_] := n!/Quotient[n, 2]!^2; t[n_] := Sum[Binomial[n, k]*sf[k], {k, 0, n}]; Table[t[n], {n,0,50}] (* _G. C. Greubel_, Aug 06 2017 *)
%o A163865 (PARI) x='x+O('x^50); Vec((1-x-4*x^2)/((1+x)*(1-3*x))^(3/2)) \\ _G. C. Greubel_, Aug 06 2017
%Y A163865 Cf. A056040, A000522, A163650.
%K A163865 nonn
%O A163865 0,2
%A A163865 _Peter Luschny_, Aug 06 2009