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A334165
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!).
2
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
OFFSET
0,3
COMMENTS
Square array of Dowling numbers.
LINKS
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.
FORMULA
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).
EXAMPLE
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, ...
MATHEMATICA
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
CROSSREFS
Columns k=1..10 give A000110 (for n > 0), A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582.
Cf. A241578, A241579, A334162 (diagonal).
Sequence in context: A144224 A122881 A210217 * A370887 A210223 A361600
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Apr 17 2020
STATUS
approved