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A293133
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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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4
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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
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refs;
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OFFSET
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0,9
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LINKS
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FORMULA
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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.
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EXAMPLE
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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, ...
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PROG
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(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
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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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