A105928 - OEIS

%I #32 Jun 01 2018 01:56:12

%S 0,0,0,6,72,780,8520,97650,1189104,15441048,213816240,3152287710,

%T 49369524600,819340272036,14373198453432,265869427695690,

%U 5173710021214560,105683257864542000,2261482144869433824,50598160483438733238,1181568482279829616680,28750554997809594831420

%N a(n) = ((n^3 - 4n + 1)*A000166(n) + (-1)^(n+1)*(n-1)^2) / 6.

%C Wang, Miska, & Mező call these 3-derangement numbers. a(n) counts the fixed point free permutations (derangements) on n + 3 elements such that the first 3 elements belong to distinct cycles. - _Istvan Mezo_, Apr 05 2017

%D P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 108.

%H Vincenzo Librandi, <a href="/A105928/b105928.txt">Table of n, a(n) for n = 0..200</a>

%H C.-Y. Wang, P. Miska, I. Mező, <a href="http://doi.org/10.1016/j.disc.2016.10.012">The r-derangement numbers</a>, Discrete Mathematics 340.7 (2017): 1681-1692.

%F G.f.: (2*x-1)*hypergeom([1,2],[],x/(1+x))/(3*(1+x)^2) - (5*x-1)*hypergeom([2,3],[],x/(1+x))/(3*(1+x)^3). - _Mark van Hoeij_, Nov 19 2011

%F E.g.f.: x^3*exp(-x)/(1-x)^4. - _Istvan Mezo_, Apr 05 2017

%t Table[((n^3 - 4 n + 1) Subfactorial[n] + (-1)^(n + 1) (n - 1)^2)/ 6, {n, 0, 21}] (* _Michael De Vlieger_, Apr 05 2017 *)

%o (PARI) s(n) = if( n<1, 1, n * s(n-1) + (-1)^n);

%o a(n) = ((n^3 - 4*n + 1) * s(n) + (-1)^(n + 1) * (n - 1)^2)/6; \\ _Indranil Ghosh_, Apr 06 2017

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Apr 27 2005