A293120 -id:A293120 - cp's OEIS frontend

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)).

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.

A293121 Expansion of e.g.f. exp(x^3/(1+x)).

Original entry on oeis.org

1, 0, 0, 6, -24, 120, -360, 0, 20160, -302400, 3628800, -39916800, 419126400, -4151347200, 36324288000, -207048441600, -1743565824000, 103742166528000, -2925529096089600, 69945932735078400, -1571249213614080000, 34354603773794304000, -741528257908838400000

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

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

Column k=2 of A293133.
Cf. A111884, A293120.
Cf. A293118.

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))))

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

a(n) = (-1)^n * A293118(n).

cp's OEIS Frontend

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)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Ruby

Formula