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A334165
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Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (k*j + 1)^n / (k^j * j!).
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2
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1, 1, 2, 1, 2, 5, 1, 2, 6, 15, 1, 2, 7, 24, 52, 1, 2, 8, 35, 116, 203, 1, 2, 9, 48, 214, 648, 877, 1, 2, 10, 63, 352, 1523, 4088, 4140, 1, 2, 11, 80, 536, 3008, 12349, 28640, 21147, 1, 2, 12, 99, 772, 5307, 29440, 112052, 219920, 115975, 1, 2, 13, 120, 1066, 8648, 60389, 324096, 1120849, 1832224, 678570
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OFFSET
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0,3
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COMMENTS
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Square array of Dowling numbers.
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LINKS
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FORMULA
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G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).
E.g.f. of column k: exp(x + (exp(k*x) - 1) / k).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
5, 6, 7, 8, 9, 10, ...
15, 24, 35, 48, 63, 80, ...
52, 116, 214, 352, 536, 772, ...
203, 648, 1523, 3008, 5307, 8648, ...
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (Exp[k x] - 1)/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
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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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