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

1, 2, 5, 16, 47, 146, 447, 1380, 4251, 13102, 40343, 124136, 381625, 1172198, 3597401, 11031012, 33798339, 103477590, 316581567, 967900224, 2957316429, 9030317478, 27558851565, 84059345244, 256265811333, 780885245826, 2378410969977, 7241027262280

Offset: 0

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn

Comments

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).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
• Peter Luschny, Swinging Factorial.

Crossrefs

Cf. A056040, A000522, A163650.

Programs

Maple

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

Mathematica

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 *)
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 *)

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

Formula

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

O.g.f.: (1-x-4*x^2)/((1+x)*(1-3*x))^(3/2). - Peter Luschny, Oct 31 2013

a(n) ~ 3^(n - 1/2) * sqrt(n) / (2*sqrt(Pi)). - Vaclav Kotesovec, Nov 27 2017