This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A070779 #49 Feb 21 2025 12:32:51
%S A070779 1,5,28,185,1426,12607,125882,1401409,17209234,231033431,3365440882,
%T A070779 52855452817,890097287834,15996379554079,305519496498106,
%U A070779 6178746162639617,131885301216119842,2962568890205560999,69853182607494217154,1724761580035969997521,44501146220521229674282
%N A070779 Expansion of e.g.f.: (exp(x/(1-x))*(2-x)-1+x)/(1-x)^3.
%C A070779 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 A070779 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 A070779 Robert Israel, <a href="/A070779/b070779.txt">Table of n, a(n) for n = 0..442</a>
%H A070779 Mark Colarusso and Sam Evens, <a href="https://arxiv.org/abs/2502.10912">Orbits on a product of two flags and a line and the Bruhat Order, I</a>, arXiv:2502.10912 [math.RT], 2025. See p. 16.
%F A070779 In Maple notation, a(n) = n! *(n+1)^2 *hypergeom([1, -n], [2, 2], -1).
%F A070779 a(n) = (n+1)!*(LaguerreL(n+1, -1)-1). - _Vladeta Jovovic_, Oct 24 2003
%F A070779 a(n) = A002720(n) - A000142(n) = Sum_{k=0..n-1} k!*binomial(n,k)^2. - _James East_, May 03 2007
%F A070779 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 A070779 a(n) = Sum_{k=0..n} A355266(n+1, k+1). - _Mélika Tebni_, Jul 07 2022
%p A070779 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 A070779 map(f, [$0..30]); # _Robert Israel_, Jan 03 2019
%p A070779 # alternative
%p A070779 A070779 := proc(n)
%p A070779 n!*(n+1)^2*hypergeom([1,-n],[2,2],-1) ;
%p A070779 simplify(%) ;
%p A070779 end proc: # _R. J. Mathar_, Jul 16 2020
%t A070779 Table[(n + 1)! (LaguerreL[n + 1, -1] -1), {n, 0, 20}] (* _Vincenzo Librandi_, Jan 04 2019 *)
%t A070779 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 A070779 (Sage)
%o A070779 @cached_function
%o A070779 def a(n):
%o A070779 if n < 3: return [1, 5, 28][n]
%o A070779 return n*(n-1)^2*a(n-3)-3*n^2*a(n-2)+(3*n+2)*a(n-1)
%o A070779 [a(n) for n in (0..20)] # _Peter Luschny_, Jan 04 2019
%Y A070779 Cf. A002720, A000142, A355266.
%K A070779 nonn
%O A070779 0,2
%A A070779 _Karol A. Penson_, May 06 2002
%E A070779 New description from _Vladeta Jovovic_, Apr 10 2003
%E A070779 Edited by _Robert Israel_, Jan 03 2019
%E A070779 Definition clarified by _Harvey P. Dale_, Sep 07 2020