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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.)