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A351776
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.
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5
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1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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E.g.f. of column k: 1/(1 + k*x*exp(x)).
T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, -1, -2, -3, -4, -5, ...
0, 0, 4, 12, 24, 40, ...
0, 3, -6, -63, -204, -465, ...
0, -4, -8, 420, 2288, 7180, ...
0, -25, 150, -3435, -32020, -138525, ...
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PROG
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(PARI) T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);
(PARI) T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
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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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