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A291709
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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(Sum_{j>=1} (-1)^(j-1)*binomial(-k,j-1)*x^j/j).
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8
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 73, 120, 1, 1, 1, 6, 33, 154, 501, 720, 1, 1, 1, 7, 46, 273, 1306, 4051, 5040, 1, 1, 1, 8, 61, 436, 2721, 12976, 37633, 40320, 1, 1, 1, 9, 78, 649, 4956, 31701, 147484, 394353, 362880, 1
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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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Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.
A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} B(j,k)*A(n-j,k)/(n-j)! for n > 0.
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EXAMPLE
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Square array B(j,k) begins:
1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, ...
0, 1, 3, 6, 10, ...
0, 1, 4, 10, 20, ...
0, 1, 5, 15, 35, ...
0, 1, 6, 21, 56, ...
Square array A(n,k) begins:
1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, ...
1, 6, 13, 22, 33, ...
1, 24, 73, 154, 273, ...
1, 120, 501, 1306, 2721, ...
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MATHEMATICA
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B[j_, k_] := (-1)^(j-1)*Binomial[-k, j-1];
A[0, _] = 1; A[n_, k_] := (n-1)!*Sum[B[j, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
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CROSSREFS
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Columns k=0..10 give A000012, A000142, A000262, A049376, A049377, A049378, A049402, A132164, A293986, A293987, A293988.
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KEYWORD
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AUTHOR
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STATUS
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approved
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