A292908 -id:A292908 - cp's OEIS frontend

A292889 Expansion of e.g.f. exp(x^3 * exp(x)).

1, 0, 0, 6, 24, 60, 480, 5250, 40656, 302904, 3024720, 35260830, 400499880, 4619610996, 58758862344, 813678468330, 11686918766880, 173172973038960, 2695921067029536, 44219170187746614, 755363769931590840, 13351503268881437100, 244634794020399856920

Offset: 0

Seiichi Manyama, Sep 26 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..498

Column k=3 of A292978.

Cf. A145453, A292891, A292908.

x='x+O('x^66); Vec(serlaplace(exp(x^3*exp(x))))

a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(k!*(n-3*k)!)); \\ Seiichi Manyama, Jul 10 2022

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k)/(k! * (n-3*k)!). - Seiichi Manyama, Jul 10 2022

A292973 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (-1)^(k+1) * k! * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, 1, 2, 1, 0, -1, -5, 1, 0, 2, -2, 15, 1, 0, 0, -6, 9, -52, 1, 0, 0, 6, 24, -4, 203, 1, 0, 0, 0, -24, -140, -95, -877, 1, 0, 0, 0, 24, 60, 870, 414, 4140, 1, 0, 0, 0, 0, -120, 240, -5922, 49, -21147, 1, 0, 0, 0, 0, 120, 360, -4830, 45416, -10088, 115975

Offset: 0

Seiichi Manyama, Sep 27 2017

			Square array begins:
   1,  1,  1,   1,  1, ...
  -1,  1,  0,   0,  0, ...
   2, -1,  2,   0,  0, ...
  -5, -2, -6,   6,  0, ...
  15,  9, 24, -24, 24, ...

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

Columns k=0-5 give: A292935, A003725, A292907, A292908, A292969, A292970.
Rows n=0 gives A000012.
Main diagonal gives A000142.
Cf. A292948, A292978.

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
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) * f(k) * (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 A292973(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 A292973(20)

T(n,k) = n! * Sum_{j=0..floor(n/k)} (-j)^(n-k*j)/(j! * (n-k*j)!) for k > 0. - Seiichi Manyama, Jul 10 2022

A292910 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 26 2017

sign

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

Column k=3 of A292948.

Cf. A145453, A292908.

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

A293017 E.g.f.: exp(-x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, -6, -24, -60, 240, 4830, 39984, 180936, -605520, -24616350, -318005160, -2385790836, -86488584, 350543790870, 6917020827360, 79778558317200, 357117438258144, -9237998478286134, -304182278908538040, -5166739059890655660, -48968796671246843160

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..505

Column k=3 of A293019.

Cf. A292908.

seq(k!*coeftayl(exp(-x^3*exp(x)), x = 0, k),k=0..30); # Muniru A Asiru, Oct 01 2017

With[{nn=30},CoefficientList[Series[Exp[-x^3Exp[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 04 2019 *)

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

a(n) = (-1)^n * A292908(n).

cp's OEIS Frontend

A292889 Expansion of e.g.f. exp(x^3 * exp(x)).

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A292973 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (-1)^(k+1) * k! * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

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A292910 E.g.f.: exp(1/3! * x^3 * exp(-x)).

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A293017 E.g.f.: exp(-x^3 * exp(x)).

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