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%I #28 Jul 27 2021 04:04:29
%S 9,53,181,465,1001,1909,3333,5441,8425,12501,17909,24913,33801,44885,
%T 58501,75009,94793,118261,145845,178001,215209,257973,306821,362305,
%U 425001,495509,574453,662481,760265,868501,987909,1119233,1263241
%N a(n) = (1/n!)*A001688(n).
%C Number of injections from {1,2,3,4} to {1,2,...,n} with no fixed points. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006
%C In general (cf. A094792, A094794, A094795, etc.), the number of injections [k] -> [n] with no fixed points is given by Sum_{i=0..k} (-1)^i*binomial(k,i)*(n-i)!/(n-k)!, which is equal to (1/n!)*f_k(n) where f_k(n) gives the k-th differences of factorial numbers. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = n^4 + 6*n^3 + 17*n^2 + 20*n + 9.
%F a(n) = Sum_{i=0..4} (-1)^i*binomial(4,i)*(n-i)!/(n-4)!. - Fiona T. Brunk (fbrunk(AT)mcs.st-and.ac.uk), May 23 2006
%F G.f.: -(x^4+6*x^2+8*x+9) / (x-1)^5. - _Colin Barker_, Jun 16 2013
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Fung Lam_, Apr 17 2014
%F P-recursive: n*a(n) = (n+5)*a(n-1) - a(n-2) with a(0) = 9 and a(1) = 53. Cf. A094791. - _Peter Bala_, Jul 25 2021
%t LinearRecurrence[{5,-10,10,-5,1},{9,53,181,465,1001},40] (* _Harvey P. Dale_, May 23 2016 *)
%Y Cf. A001563, A001564, A001565, A001688, A001689, A023043.
%Y Cf. A094791, A094792, A094794, A094795.
%K nonn,easy
%O 0,1
%A _Benoit Cloitre_, Jun 11 2004
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