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A361781
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A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.
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3
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1, 1, 1, 1, 0, 2, 1, -1, 1, 5, 1, -2, 2, 1, 15, 1, -3, 5, -3, 4, 52, 1, -4, 10, -13, 7, 11, 203, 1, -5, 17, -35, 36, -10, 41, 877, 1, -6, 26, -75, 127, -101, 31, 162, 4140, 1, -7, 37, -139, 340, -472, 293, -21, 715, 21147, 1, -8, 50, -233, 759, -1573, 1787, -848, 204, 3425, 115975
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OFFSET
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0,6
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LINKS
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FORMULA
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E.g.f. of column k: exp(exp(x) - k*x - 1).
A(n,k) = Sum_{j=0..n} (-k)^j*binomial(n,j)*Bell(n-j).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, -1, -2, -3, -4, -5, -6, ...
2, 1, 2, 5, 10, 17, 26, 37, ...
5, 1, -3, -13, -35, -75, -139, -233, ...
15, 4, 7, 36, 127, 340, 759, 1492, ...
52, 11, -10, -101, -472, -1573, -4214, -9685, ...
203, 41, 31, 293, 1787, 7393, 23711, 63581, ...
877, 162, -21, -848, -6855, -35178, -134873, -421356, ...
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MAPLE
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A:= proc(n, k) option remember; uses combinat;
add(binomial(n, j)*(-k)^j*bell(n-j), j=0..n)
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, m) option remember;
`if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
end:
A:= (n, k)-> b(n, -k):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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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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