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 A082491 a(n) = n! * d(n), where n! = factorial numbers (A000142), d(n) = subfactorial numbers (A000166). 9

%I

%S 1,0,2,12,216,5280,190800,9344160,598066560,48443028480,4844306476800,

%T 586161043776000,84407190782745600,14264815236056985600,

%U 2795903786354347468800,629078351928420506112000

%N a(n) = n! * d(n), where n! = factorial numbers (A000142), d(n) = subfactorial numbers (A000166).

%C a(n) is also the number of pairs of n-permutations p and q such that p(x)<>q(x) for each x in { 1, 2, ..., n }.

%C Or number of nXn matrices with exactly one 1 and one 2 in each row and column, other entries 0 (cf. A001499). [From _Vladimir Shevelev_, Mar 22 2010]

%H Vincenzo Librandi, <a href="/A082491/b082491.txt">Table of n, a(n) for n = 0..200</a>

%H Ira Gessel, <a href="http://www.mat.univie.ac.at/~slc/opapers/s17gessel.html">Enumerative applications of symmetric functions</a>

%F a(n) = n! * d(n) where d(n) = A000166(n).

%F a(n) = sum(k=0..n, binomial(n, k)^2 * (-1)^k * (n - k)!^2 * k! ).

%F a(n+2) = (n+2)*(n+1) * ( a(n+1) + (n+1)*a(n) ).

%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=0..15); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 2008

%t Table[Subfactorial[n]*n!, {n, 0, 15}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

%o (Maxima) A000166[0]:1\$

%o A000166[n]:=n*A000166[n-1]+(-1)^n\$

%o makelist(n!*A000166[n], n, 0, 12); /* Emanuele Munarini, Mar 01 2011 */

%o (PARI)

%o d(n)=if(n<1, n==0, n*d(n-1)+(-1)^n);

%o a(n)=d(n)*n!;

%o vector(33,n,a(n-1))

%o /* _Joerg Arndt_, May 28 2012 */

%Y Cf. A000142, A000166.

%K easy,nonn

%O 0,3

%A _Emanuele Munarini_, Apr 28 2003

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