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

0,2

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A056040, A000522, A163650.

Sequence in context: A148376 A148377 A349270 * A148378 A148379 A257970

Adjacent sequences: A163862 A163863 A163864 * A163866 A163867 A163868

Keyword

nonn

Author

Peter Luschny, Aug 06 2009

Status

approved