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

0,4

Comments

Analog to the subfactorial A000166.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Peter Luschny, Swinging Factorial.

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

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 *)

Prog

(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

Crossrefs

Row sums of A163649. Cf. A056040, A000166.

Sequence in context: A272199 A260074 A294270 * A259777 A365038 A013981

Adjacent sequences: A163647 A163648 A163649 * A163651 A163652 A163653

Keyword

sign

Author

Peter Luschny, Aug 02 2009

Status

approved