A113059 a(n) = n! * Sum_{k=0..n} A000296(k)/k!.

1, 1, 3, 10, 44, 231, 1427, 10151, 81923, 740732, 7425042, 81773715, 981864897, 12767876941, 178774288331, 2681781213130, 42909715480460, 729474427239587, 13130613291110603, 249482261007109579, 4989650444408388515, 104782705832468197252, 2305219956684224457858

Offset: 0

Author

Karol A. Penson, Oct 12 2005

Keywords

• nonn

Comments

Number of set partitions of [n] where the k-th singletons are k-colored and all other blocks are unicolored. - Alois P. Heinz, Apr 29 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A000110, A000142, A000255, A000296, A101053, A101054.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)-1-x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, k*b(n-1, k+1)+
      add(b(n-j, k)*binomial(n-1, j-1), j=2..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 29 2025
# second Maple program:
b:= proc(n, k, m) option remember; `if`(n=0, k!, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 29 2025

Mathematica

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x] - 1 - x]/(1 - x), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, May 23 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace( exp(exp(x)-1-x)/(1-x))) \\ G. C. Greubel, May 23 2018

Formula

a(n) = (-1)^n*n!*Sum_{k >=0} LaguerreL(n, -n-1, k-1)/k!/exp(1), n>=0.

E.g.f.: exp(exp(x)-1-x)/(1-x).

a(n) ~ exp(exp(1)-2) * n!. - Vaclav Kotesovec, Jun 26 2022