%I #46 Jul 07 2022 10:46:42
%S 1,5,28,185,1426,12607,125882,1401409,17209234,231033431,3365440882,
%T 52855452817,890097287834,15996379554079,305519496498106,
%U 6178746162639617,131885301216119842,2962568890205560999,69853182607494217154,1724761580035969997521,44501146220521229674282
%N Expansion of e.g.f.: (exp(x/(1-x))*(2-x)-1+x)/(1-x)^3.
%C Equal to the number of strictly partial permutations on [n]; i.e. equal to the cardinality of the complement I_n\S_n, where I_n and S_n denote the symmetric inverse monoid and symmetric group on [n]. - _James East_, May 03 2007
%C Former name was "E.g.f.: (exp(x/(1-x))-1)/(1-x)." However, that would be the e.g.f. with offset 1 rather than 0. - _Robert Israel_, Jan 03 2019
%H Robert Israel, <a href="/A070779/b070779.txt">Table of n, a(n) for n = 0..442</a>
%F In Maple notation, a(n) = n! *(n+1)^2 *hypergeom([1, -n], [2, 2], -1).
%F a(n) = (n+1)!*(LaguerreL(n+1, -1)-1). - _Vladeta Jovovic_, Oct 24 2003
%F a(n) = A002720(n) - A000142(n) = Sum_{k=0..n-1} k!*binomial(n,k)^2. - _James East_, May 03 2007
%F D-finite with recurrence a(n) = (3*n+2)*a(n-1) - 3*n^2*a(n-2) + n*(n-1)^2*a(n-3). - _Robert Israel_, Jan 03 2019
%F a(n) = Sum_{k=0..n} A355266(n+1, k+1). - _Mélika Tebni_, Jul 07 2022
%p f:= gfun:-rectoproc({(n + 3)*(n + 2)^2*a(n) - 3*(n + 3)^2*a(n + 1) + (3*n + 11)*a(n + 2) - a(n + 3)=0, a(0)=1,a(1)=5,a(2)=28},a(n),remember):
%p map(f, [$0..30]); # _Robert Israel_, Jan 03 2019
%p # alternative
%p A070779 := proc(n)
%p n!*(n+1)^2*hypergeom([1,-n],[2,2],-1) ;
%p simplify(%) ;
%p end proc: # _R. J. Mathar_, Jul 16 2020
%t Table[(n + 1)! (LaguerreL[n + 1, -1] -1), {n, 0, 20}] (* _Vincenzo Librandi_, Jan 04 2019 *)
%t With[{nn=20},CoefficientList[Series[(Exp[x/(1-x)](2-x)-1+x)/(1-x)^3,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Sep 07 2020 *)
%o (Sage)
%o @cached_function
%o def a(n):
%o if n < 3: return [1, 5, 28][n]
%o return n*(n-1)^2*a(n-3)-3*n^2*a(n-2)+(3*n+2)*a(n-1)
%o [a(n) for n in (0..20)] # _Peter Luschny_, Jan 04 2019
%Y Cf. A002720, A000142, A355266.
%K nonn
%O 0,2
%A _Karol A. Penson_, May 06 2002
%E New description from _Vladeta Jovovic_, Apr 10 2003
%E Edited by _Robert Israel_, Jan 03 2019
%E Definition clarified by _Harvey P. Dale_, Sep 07 2020