A292866 a(n) = n! * [x^n] exp(n*(1 - exp(x))).

1, -1, 2, -3, -20, 370, -4074, 34293, -138312, -2932533, 106271090, -2192834490, 32208497124, -206343936097, -7657279887698, 412496622532785, -12455477719752976, 260294034150380430, -2256541295745391542, -122593550603339550843, 8728842979656718306780

Offset: 0

Author

Seiichi Manyama, Sep 25 2017

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..415

Crossrefs

Main diagonal of A292861.

Cf. A242817.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      -(1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 25 2017

Mathematica

Table[n!*SeriesCoefficient[E^(n*(1 - E^x)),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 25 2017 *)
a[n_] := BellB[n, -n]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 23 2021 *)

PARI

{a(n) = sum(k=0, n, (-n)^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019

Ruby

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << k * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[j]}}
  ary
end
def A292866(n)
  (0..n).map{|i| A(-i, i)[-1]}
end
p A292866(20)

Formula

a(n) = exp(n) * Sum_{k>=0} (-n)^k*k^n/k!. - Ilya Gutkovskiy, Jul 13 2019

a(n) = Sum_{k=0..n} (-n)^k * Stirling2(n,k). - Seiichi Manyama, Jul 28 2019

a(n) = BellPolynomial(n, -n). - Peter Luschny, Dec 23 2021