Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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