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%I N0672 #44 Dec 20 2022 01:36:12
%S 0,2,6,36,220,1590,12978,118664,1201464,13349610,161530270,2114578092,
%T 29780308116,448995414686,7215997736010,123153028027920,
%U 2224451568754288,42395429898611154,850263899633257014,17900292623858042420,394701452356069835340,9096928711444657157382,218739785834282892557026
%N Let u(1)=0, u(2)=1, u(k)=u(k-1)+u(k-2)/(k-2); then a(n)=n!*u(n).
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263, Table 7.5.1, row 3.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210, Table 3, Three-line Latin rectangles.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H Vincenzo Librandi, <a href="/A086325/b086325.txt">Table of n, a(n) for n = 1..200</a>
%H Gerzson Kéri, <a href="http://ac.inf.elte.hu/Vol_053_2022/093_53.pdf">The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas</a>, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.
%F a(n) = ceiling(n*n!/e) - (1-(-1)^n)/2.
%F E.g.f.: x^2*exp(-x)/(1-x)^2. - _Vladeta Jovovic_, Nov 20 2003
%F a(n) = n*floor((n!+1)/e). [_Gary Detlefs_, Jul 13 2010]
%F a(n) = n * A000166(n). [_Joerg Arndt_, Jul 09 2012]
%F G.f.: x*f'(x), where f(x) = 1/(1 + x) + Sum_{k>=1} k^k*x^k/(1 + (k + 1)*x)^(k+1). - _Ilya Gutkovskiy_, Apr 13 2017
%p a:=n->n!*add((-1)^k/k!, k=0..n): seq(a(n)*n, n=1..19); # _Zerinvary Lajos_, Dec 18 2007
%p with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*fibonacci(2,n), n=1..19); # _Zerinvary Lajos_, Jun 11 2008
%p with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*n, n=1..19); # _Zerinvary Lajos_, Jun 11 2008
%t Table[Subfactorial[n]*n, {n, 1, 19}] (* _Zerinvary Lajos_, Jul 09 2009 *)
%o (PARI) a(n) = n*((n! + 1)\exp(1)); \\ _Indranil Ghosh_, Apr 13 2017
%Y Cf. A000246, A000274, A000166.
%K nonn
%O 1,2
%A _N. J. A. Sloane_. This sequence appeared in the 1973 "Handbook", but was then omitted from the database. Resubmitted by _Benoit Cloitre_, Aug 30 2003. Entry revised by _N. J. A. Sloane_, Jun 11 2012
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