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!).

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

Author

Ilya Gutkovskiy, Apr 17 2020

Keywords

• nonn
• tabl

Comments

Square array of Dowling numbers.

Examples

			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,  ...
		

Links

• Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Crossrefs

Columns k=1..10 give A000110 (for n > 0), A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582.

Cf. A241578, A241579, A334162 (diagonal).

Programs

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

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).