A163650 Subswing - the inverse binomial transform of the swinging factorial (A056040).

1, 0, 1, 2, -9, 44, -165, 594, -2037, 6824, -22437, 72830, -234047, 746316, -2364947, 7455798, -23405085, 73207728, -228275949, 709906518, -2202557691, 6819616020, -21076580511, 65032888998, -200369138571, 616531573224, -1894784517675, 5816886949874

Offset: 0

Author

Peter Luschny, Aug 02 2009

Keywords

• sign
• changed

Comments

Analog to the subfactorial A000166.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Luschny, Swinging Factorial.

Crossrefs

Row sums of A163649. Cf. A056040, A000166.

Programs

Maple

a := proc(n) local k: add((-1)^(n-k)*binomial(n,k)*(k!/iquo(k,2)!^2), k=0..n) end:

Mathematica

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 *)

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

Formula

E.g.f.: exp(-x)*BesselI(0,2*x)*(1+x). - Peter Luschny, Aug 26 2012

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)*(k!/(floor(k/2)!)^2). - G. C. Greubel, Aug 01 2017

a(n) ~ -(-1)^n * sqrt(n) * 3^(n - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Oct 31 2017

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