A292910 -id:A292910 - 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

A145453 Exponential transform of binomial(n,3) = A000292(n-2).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 30, 175, 1176, 7084, 42120, 286605, 2270180, 19213766, 166326524, 1497096055, 14374680880, 147259920760, 1582837679056, 17659771122969, 204674606377140, 2473357218561250, 31148510170120420, 407154732691440811, 5504706823227724904

Offset: 0

Alois P. Heinz, Oct 10 2008

nonn

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 3 balls are seen at the top.

a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 3 labels.

Seiichi Manyama, Table of n, a(n) for n = 0..530 (terms 0..200 from Alois P. Heinz)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

3rd column of A145460, A143398.

Cf. A292889, A292910.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,3) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

Table[Sum[BellY[n, k, Binomial[Range[n], 3]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

E.g.f.: exp(exp(x)*x^3/3!).

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)

A292954 E.g.f.: exp(-1/3! * x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, -1, -4, -10, -10, 105, 1064, 6356, 25080, 9075, -1056660, -13219206, -106106364, -548948855, 139658960, 48411569800, 761039099824, 7815284148711, 52216924707660, 9385130453790, -6650556642220260, -132749143322588331, -1713641693856894824

Offset: 0

Seiichi Manyama, Sep 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..536

Column k=3 of A293015.
Cf. A292952 (k=1), A292953 (k=2), this sequence (k=3), A292955 (k=4).
Cf. A292910.

x='x+O('x^66); Vec(serlaplace(exp(-1/3!*x^3*exp(x))))

a(n) = (-1)^n * A292910(n).

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

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A145453 Exponential transform of binomial(n,3) = A000292(n-2).

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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.

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A292954 E.g.f.: exp(-1/3! * x^3 * exp(x)).

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