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 A163865 The binomial transform of the swinging factorial (A056040). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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). REFERENCES Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008. 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 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: A075887 A148376 A148377 * A148378 A148379 A257970 Adjacent sequences:  A163862 A163863 A163864 * A163866 A163867 A163868 KEYWORD nonn AUTHOR Peter Luschny, Aug 06 2009 STATUS approved

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Last modified October 17 18:19 EDT 2018. Contains 316292 sequences. (Running on oeis4.)