A292912 -id:A292912 - cp's OEIS frontend

A003725 E.g.f.: exp( x * exp(-x) ).

1, 1, -1, -2, 9, -4, -95, 414, 49, -10088, 55521, -13870, -2024759, 15787188, -28612415, -616876274, 7476967905, -32522642896, -209513308607, 4924388011050, -38993940088199, 11731860520780, 3807154270837281

Offset: 0

R. H. Hardin

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..557
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3.

Cf. A000248, A069856, A072034. A215652.

Cf. this sequence (k=1), A292909 (k=2), A292910 (k=3), A292912 (k=4).

With[{nn=30},CoefficientList[Series[Exp[x Exp[-x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Oct 20 2011 *)

Vec(serlaplace(exp(exp(-x) * x))) \\ Charles R Greathouse IV, Sep 26 2017

a(n) = Sum_{k=0..n} (-k)^(n-k)*binomial(n, k). - Vladeta Jovovic, Mar 15 2003
First column of A215652. - Peter Bala, Sep 14 2012
G.f.: Sum_{k>=0} x^k/(1 + k*x)^(k+1). - Ilya Gutkovskiy, Jun 25 2018

A292948 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = (-1)^(k+1) * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, 1, 2, 1, 0, -1, -5, 1, 0, 1, -2, 15, 1, 0, 0, -3, 9, -52, 1, 0, 0, 1, 9, -4, 203, 1, 0, 0, 0, -4, -40, -95, -877, 1, 0, 0, 0, 1, 10, 210, 414, 4140, 1, 0, 0, 0, 0, -5, -10, -1176, 49, -21147, 1, 0, 0, 0, 0, 1, 15, -105, 7273, -10088, 115975, 1, 0, 0

Offset: 0

Seiichi Manyama, Sep 27 2017

			Square array begins:
    1,  1,  1,  1, 1, ...
   -1,  1,  0,  0, 0, ...
    2, -1,  1,  0, 0, ...
   -5, -2, -3,  1, 0, ...
   15,  9,  9, -4, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A292935, A003725, A292909, A292910, A292912, A292950.
Rows n=0 gives A000012.
Main diagonal gives A000012.
Cf. A145460.

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 << (-1) ** (k % 2 + 1) * (0..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292948(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A292948(20)

cp's OEIS Frontend

A003725 E.g.f.: exp( x * exp(-x) ).

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