A070779 Expansion of e.g.f.: (exp(x/(1-x))*(2-x)-1+x)/(1-x)^3.

1, 5, 28, 185, 1426, 12607, 125882, 1401409, 17209234, 231033431, 3365440882, 52855452817, 890097287834, 15996379554079, 305519496498106, 6178746162639617, 131885301216119842, 2962568890205560999, 69853182607494217154, 1724761580035969997521, 44501146220521229674282

Offset: 0

Author

Karol A. Penson, May 06 2002

Keywords

• nonn

Comments

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

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

Links

• Robert Israel, Table of n, a(n) for n = 0..442
• Mark Colarusso and Sam Evens, Orbits on a product of two flags and a line and the Bruhat Order, I, arXiv:2502.10912 [math.RT], 2025. See p. 16.

Crossrefs

Cf. A002720, A000142, A355266.

Programs

Maple

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):
map(f, [$0..30]); # Robert Israel, Jan 03 2019
# alternative
A070779 := proc(n)
    n!*(n+1)^2*hypergeom([1,-n],[2,2],-1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Jul 16 2020

Mathematica

Table[(n + 1)! (LaguerreL[n + 1, -1] -1), {n, 0, 20}] (* Vincenzo Librandi, Jan 04 2019 *)
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 *)

Sage

@cached_function
def a(n):
    if n < 3: return [1, 5, 28][n]
    return n*(n-1)^2*a(n-3)-3*n^2*a(n-2)+(3*n+2)*a(n-1)
[a(n) for n in (0..20)] # Peter Luschny, Jan 04 2019

Formula

In Maple notation, a(n) = n! *(n+1)^2 *hypergeom([1, -n], [2, 2], -1).

a(n) = (n+1)!*(LaguerreL(n+1, -1)-1). - Vladeta Jovovic, Oct 24 2003

a(n) = A002720(n) - A000142(n) = Sum_{k=0..n-1} k!*binomial(n,k)^2. - James East, May 03 2007

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

a(n) = Sum_{k=0..n} A355266(n+1, k+1). - Mélika Tebni, Jul 07 2022

Extensions

New description from Vladeta Jovovic, Apr 10 2003

Edited by Robert Israel, Jan 03 2019

Definition clarified by Harvey P. Dale, Sep 07 2020