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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)). 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

FORMULA

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.

EXAMPLE

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, ...

PROG

(Ruby)

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)

CROSSREFS

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,

Sequence in context: A057516 A293015 A293119 * A178471 A160381 A089311

Adjacent sequences:  A293130 A293131 A293132 * A293134 A293135 A293136

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Sep 30 2017

STATUS

approved

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Last modified May 26 00:32 EDT 2020. Contains 334613 sequences. (Running on oeis4.)