|
%I #21 Aug 03 2021 19:01:52
%S 1,2,5,9,27,123,693,4653,36111,317583,3118617,33804177,400755267,
%T 5156954019,71572594557,1065571143093,16938122939703,286298719063863,
%U 5127206924693601,96975312507734553,1931609062232400747,40414621201681598667,886153986344092389957
%N a(n) = d(n-1) + d(n-2) + (n-1)[d(n-2) + 2d(n-3) + d(n-4)], where d(n), the derangement numbers, are given in A000166. (Let d(n) = 0 if n < 0.)
%H Y.-R. Liu and M. R. Murthy, <a href="http://dx.doi.org/10.1016/j.jcta.2004.11.004">Sieve methods in combinatorics</a>, J. Combinatorial Theory, Ser. A, 111 (2005), 1-23.
%t a[1] = 1; a[2] = 2; a[3] = 5; a[n_] := a[n] = ((2*n^3 - 27*n^2 + 115*n - 150)*a[n - 3] + (2*n^3 - 23*n^2 + 85*n - 100)*a[n - 2] + 3*(2*n - 9)*a[n - 1])/(2*n - 9); Table[a[n], {n, 1, 23}]
%t (* or: *)
%t d[n_] := If[n < 0, 0, Subfactorial[n]]; Table[(n - 1)*(d[n - 4] + 2*d[n - 3] + d[n - 2]) + d[n - 2] + d[n - 1], {n, 1, 23}](* _Jean-François Alcover_, Nov 03 2016 *)
%o (PARI) d(n)=if(n>0, n!\/exp(1), n==0)
%o a(n)=d(n-1) + d(n-2) + (n-1)*(d(n-2) + 2*d(n-3) + d(n-4)) \\ _Charles R Greathouse IV_, Nov 03 2016
%Y Cf. A000166, A109743.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Aug 13 2005
|
