A293134 -id:A293134 - cp's OEIS frontend

A293125 Expansion of e.g.f.: exp(-x/(1+x)).

1, -1, 3, -13, 73, -501, 4051, -37633, 394353, -4596553, 58941091, -824073141, 12470162233, -202976401213, 3535017524403, -65573803186921, 1290434218669921, -26846616451246353, 588633468315403843, -13564373693588558173, 327697927886085654441

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

For n >= 1, gives row sums of A008297, triangle of Lah numbers. - Daniel Forgues, Oct 12 2019

Seiichi Manyama, Table of n, a(n) for n = 0..444
Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.

Column k=0 of A293134.

Cf. A000262, A008297.

a:=[-1,3];; for n in [3..25] do a[n]:=-(2*n-1)*a[n-1]-(n-2)*(n-1)*a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Dec 04 2018

[1] cat [(-1)^n*Factorial(n)*(&+[Binomial(n-1, j)/Factorial(j+1): j in [0..n-1]]): n in [1..30]]; // G. C. Greubel, Dec 04 2018

CoefficientList[Series[E^(-x/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)
a[n_] := If[n == 0, 1, (-1)^n n! Hypergeometric1F1[1 - n, 2, -1]];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Oct 13 2019 *)

my(x='x+O('x^21)); Vec(serlaplace(exp(-x/(1+x))))

[1] + [(-1)^n*factorial(n)*sum(binomial(n-1,j)/factorial(j+1) for j in (0..n-1)) for n in (1..30)] # G. C. Greubel, Dec 04 2018

a(n) = (-1)^n * A000262(n).
From Vaclav Kotesovec, Sep 30 2017: (Start)
a(n) = -(2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2).
a(n) ~ (-1)^n * n^(n-1/4) * exp(-1/2 + 2*sqrt(n) - n) / sqrt(2) * (1 - 5/(48*sqrt(n)) - 95/(4608*n)). (End)
a(n) = (-1)^n * n! * Sum_{j=0..n-1} binomial(n-1, j)/(j+1)!, for n > 0. - G. C. Greubel, Dec 04 2018
a(n) = (-1)^n*n!*hypergeom([1 - n], [2], -1) for n > 0. - Peter Luschny, Oct 13 2019

A293133 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^(k+1)/(1+x)).

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, 0, 2, 1, 1, 0, 0, -6, 1, 1, 0, 0, 6, 36, -19, 1, 0, 0, 0, -24, -240, 151, 1, 0, 0, 0, 24, 120, 1920, -1091, 1, 0, 0, 0, 0, -120, -360, -17640, 7841, 1, 0, 0, 0, 0, 120, 720, 0, 183120, -56519, 1, 0, 0, 0, 0, 0, -720, -5040, 20160, -2116800

Offset: 0

Seiichi Manyama, Sep 30 2017

			Square array begins:
     1,    1,   1,    1, ...
     1,    0,   0,    0, ...
    -1,    2,   0,    0, ...
     1,   -6,   6,    0, ...
     1,   36, -24,   24, ...
   -19, -240, 120, -120, ...

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

Columns k=0..2 give A111884, A293120, A293121.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
A(n,n-1) gives A000142(n).
Cf. A293053, A293119, A293134,

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) * (k..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293133(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 A293133(20)

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = (-1)^k * Sum_{i=k..n-1} (-1)^i*(i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.

A293122 E.g.f.: exp(-x^2/(1+x)).

Original entry on oeis.org

1, 0, -2, 6, -12, 0, 240, -2520, 21840, -181440, 1481760, -11642400, 80498880, -311351040, -4739454720, 183437654400, -4300276780800, 88851284121600, -1754954007206400, 34107089784768000, -659574028252339200, 12724865943229440000, -244046146272658329600

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..450

Column k=1 of A293134.

Cf. A293117.

CoefficientList[Series[E^(-x^2/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(-x^2/(1+x))))

a(n) = (-1)^n * A293117(n).

A293123 E.g.f.: exp(-x^3/(1+x)).

Original entry on oeis.org

1, 0, 0, -6, 24, -120, 1080, -10080, 100800, -1149120, 14515200, -199584000, 2973801600, -47740492800, 820928908800, -15049152518400, 292919058432000, -6031865968128000, 130990787582054400, -2991455760887193600, 71659101232502784000, -1796424431562528768000

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293134.

Cf. A293049.

CoefficientList[Series[E^(-x^3/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

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

a(n) = (-1)^n * A293049(n).

a(n) ~ (-1)^n * n^(n-1/4) * exp(-5/2 + 2*sqrt(n) - n) / sqrt(2). - Vaclav Kotesovec, Sep 30 2017

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A293125 Expansion of e.g.f.: exp(-x/(1+x)).

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A293133 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^(k+1)/(1+x)).

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A293122 E.g.f.: exp(-x^2/(1+x)).

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A293123 E.g.f.: exp(-x^3/(1+x)).

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