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A293024
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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(exp(x) - Sum_{i=0..k} x^i/i!).
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9
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1, 1, 1, 1, 0, 2, 1, 0, 1, 5, 1, 0, 0, 1, 15, 1, 0, 0, 1, 4, 52, 1, 0, 0, 0, 1, 11, 203, 1, 0, 0, 0, 1, 1, 41, 877, 1, 0, 0, 0, 0, 1, 11, 162, 4140, 1, 0, 0, 0, 0, 1, 1, 36, 715, 21147, 1, 0, 0, 0, 0, 0, 1, 1, 92, 3425, 115975, 1, 0, 0, 0, 0, 0, 1, 1, 36, 491, 17722, 678570
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OFFSET
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0,6
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COMMENTS
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A(n,k) is the number of set partitions of [n] into blocks of size > k.
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LINKS
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FORMULA
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E.g.f. of column k: Product_{i>k} exp(x^i/i!).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-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, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, 0, ...
2, 1, 0, 0, 0, 0, 0, 0, ...
5, 1, 1, 0, 0, 0, 0, 0, ...
15, 4, 1, 1, 0, 0, 0, 0, ...
52, 11, 1, 1, 1, 0, 0, 0, ...
203, 41, 11, 1, 1, 1, 0, 0, ...
877, 162, 36, 1, 1, 1, 1, 0, ...
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MAPLE
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A:= proc(n, k) option remember; `if`(n=0, 1, add(
A(n-j, k)*binomial(n-1, j-1), j=1+k..n))
end:
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MATHEMATICA
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A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = Sum[A[n-j, k] Binomial[n-1, j-1], {j, k+1, n}];
A[_, _] = 0;
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PROG
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(Ruby)
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 << (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-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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