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