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%I A163650 #16 Jul 04 2023 08:51:10
%S A163650 1,0,1,2,-9,44,-165,594,-2037,6824,-22437,72830,-234047,746316,
%T A163650 -2364947,7455798,-23405085,73207728,-228275949,709906518,-2202557691,
%U A163650 6819616020,-21076580511,65032888998,-200369138571,616531573224,-1894784517675,5816886949874
%N A163650 Subswing - the inverse binomial transform of the swinging factorial (A056040).
%C A163650 Analog to the subfactorial A000166.
%H A163650 G. C. Greubel, <a href="/A163650/b163650.txt">Table of n, a(n) for n = 0..1000</a>
%H A163650 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html"> Swinging Factorial.</a>
%F A163650 E.g.f.: exp(-x)*BesselI(0,2*x)*(1+x). - _Peter Luschny_, Aug 26 2012
%F A163650 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)*(k!/(floor(k/2)!)^2). - _G. C. Greubel_, Aug 01 2017
%F A163650 a(n) ~ -(-1)^n * sqrt(n) * 3^(n - 1/2) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Oct 31 2017
%F A163650 D-finite with recurrence n*a(n) +5*(n-1)*a(n-1) +(n-4)*a(n-2) +(-13*n+23)*a(n-3) +6*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jul 04 2023
%p A163650 a := proc(n) local k: add((-1)^(n-k)*binomial(n,k)*(k!/iquo(k,2)!^2), k=0..n) end:
%t A163650 sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*sf[k], {k, 0, n}]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jun 28 2013 *)
%o A163650 (PARI) for(n=0,50, print1(sum(k=0,n, (-1)^(n-k)*binomial(n,k)*(k!/((k\2)!)^2)), ", ")) \\ _G. C. Greubel_, Aug 01 2017
%Y A163650 Row sums of A163649. Cf. A056040, A000166.
%K A163650 sign
%O A163650 0,4
%A A163650 _Peter Luschny_, Aug 02 2009

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