A334192 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, 0, 1, 0, -1, 1, 0, -2, -1, 1, 0, -3, -4, 2, 1, 0, -4, -9, 4, 9, 1, 0, -5, -16, 0, 64, 9, 1, 0, -6, -25, -16, 189, 248, -50, 1, 0, -7, -36, -50, 384, 1377, 48, -267, 1, 0, -8, -49, -108, 625, 4416, 4374, -6512, -413, 1, 0, -9, -64, -196, 864, 10625, 26368, -26001, -51200, 2180

Offset: 0

Author

Ilya Gutkovskiy, Apr 18 2020

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,   1,    1,    1,    1,    1,  ...
   0,   0,    0,    0,    0,    0,  ...
  -1,  -2,   -3,   -4,   -5,   -6,  ...
  -1,  -4,   -9,  -16,  -25,  -36,  ...
   2,   4,    0,  -16,  -50, -108,  ...
   9,  64,  189,  384,  625,  864,  ...
		

Crossrefs

Columns k=1..3 give A293037, A334190, A334191.

Cf. A309386, A334165, A334193 (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 + (1 - Exp[k x])/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 + (1 - exp(k*x)) / k).