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%I #43 Nov 11 2023 14:23:12
%S 1,1,2,8,42,258,1824,14664,132360,1326120,14606640,175448160,
%T 2282469840,31972303440,479793807360,7679384173440,130586660507520,
%U 2351111258805120,44679858911251200,893744703503769600,18771276190401504000,413017883356110278400
%N From applying the "rational mean" to the number e.
%C Binomial transform of A000271. - _Vladeta Jovovic_, Jun 26 2007
%C Conjecture: this is also the number of acyclic orientations of the complement of the path graph. - _Martin Rubey_, Oct 15 2023
%H Vincenzo Librandi, <a href="/A013999/b013999.txt">Table of n, a(n) for n = 0..200</a>
%H Domingo Gómez Morín, <a href="https://web.archive.org/web/20000304112937/http://www.etheron.net/usuarios/dgomez/racional.htm">New Elements For The Irrational Numbers</a>, Journal of Transfigural Mathematics, Vol. 2, No. 1, 1996.
%H R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/multfree.pdf">An Equivalence Relation on the Symmetric Group and Multiplicity-free Flag h-Vectors</a>, preprint, 2008. - From _N. J. A. Sloane_, May 06 2012
%F G.f.: Sum_{n>=0} n!*(x*(1-x))^n. - _Vladeta Jovovic_, Jun 26 2007
%F Recurrence: a(n) = (n+3)*a(n-1) - (2*n+1)*a(n-2) + n*a(n-3). - _Vaclav Kotesovec_, Oct 07 2012
%F G.f.: 1/Q(0), where Q(k)= 1 + x/(1-x) - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 21 2013
%F a(n) = sum(binomial(n-k+1,k)*(-1)^k*(n-k+1)!, k=0..floor((n+1)/2)). - _Emanuele Munarini_, Jul 01 2013
%F a(n) ~ n!*n/exp(1). - _Vaclav Kotesovec_, Jul 06 2013
%t Table[SeriesCoefficient[Sum[k!*(x*(1-x))^k,{k,0,n}],{x,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)
%o (Maxima) makelist(sum(binomial(n-k+1,k)*(-1)^k*(n-k+1)!,k,0,floor((n+1)/2)),n,0,20); /* _Emanuele Munarini_, Jul 01 2013 */
%Y Cf. A000271.
%K nonn
%O 0,3
%A Domingo Gomez Morin (Dgomezm(AT)etheron.net)
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