A362019 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * Sum_{j=0..n} (-k*j)^j * binomial(n,j).

1, 1, -1, 1, 0, 1, 1, 1, 3, -1, 1, 2, 13, 17, 1, 1, 3, 31, 173, 169, -1, 1, 4, 57, 629, 3321, 2079, 1, 1, 5, 91, 1547, 18025, 81529, 31261, -1, 1, 6, 133, 3089, 58993, 662639, 2443333, 554483, 1, 1, 7, 183, 5417, 147081, 2888979, 29752957, 86475493, 11336753, -1

Offset: 0

Author

Seiichi Manyama, May 05 2023

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,    1,     1,      1,       1,       1, ...
  -1,    0,     1,      2,       3,       4, ...
   1,    3,    13,     31,      57,      91, ...
  -1,   17,   173,    629,    1547,    3089, ...
   1,  169,  3321,  18025,   58993,  147081, ...
  -1, 2079, 81529, 662639, 2888979, 8998399, ...
		

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Columns k=0..3 give A033999, (-1)^n * A069856(n), A362859, A362860.

Main diagonal gives A362862.

Cf. A362856.

Programs

PARI

T(n, k) = (-1)^n*sum(j=0, n, (-k*j)^j*binomial(n, j));

Formula

E.g.f. of column k: exp(-x) / (1 + LambertW(-k*x)).

G.f. of column k: Sum_{j>=0} (k*j*x)^j / (1 + x)^(j+1).