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

0,2

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

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

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 *)

Prog

(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

Crossrefs

Cf. A002720, A000142, A355266.

Sequence in context: A156629 A331797 A123776 * A189487 A367049 A024065

Adjacent sequences: A070776 A070777 A070778 * A070780 A070781 A070782

Keyword

nonn

Author

Karol A. Penson, May 06 2002

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

Status

approved