A362856 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-k)^(n-j) * j^j * binomial(n,j).

1, 1, 1, 1, 0, 4, 1, -1, 3, 27, 1, -2, 4, 17, 256, 1, -3, 7, 7, 169, 3125, 1, -4, 12, -9, 120, 2079, 46656, 1, -5, 19, -37, 121, 1373, 31261, 823543, 1, -6, 28, -83, 208, 797, 21028, 554483, 16777216, 1, -7, 39, -153, 441, 21, 14517, 373931, 11336753, 387420489

Offset

0,6

Links

Table of n, a(n) for n=0..54.

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

Formula

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

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

Example

Square array begins:
     1,    1,    1,   1,   1,     1, ...
     1,    0,   -1,  -2,  -3,    -4, ...
     4,    3,    4,   7,  12,    19, ...
    27,   17,    7,  -9, -37,   -83, ...
   256,  169,  120, 121, 208,   441, ...
  3125, 2079, 1373, 797,  21, -1525, ...

Prog

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

Crossrefs

Columns k=0..3 give A000312, (-1)^n * A069856(n), A362857, A362858.

Main diagonal gives A290158.

Cf. A362019.

Sequence in context: A255511 A014518 A316584 * A146325 A333807 A069289

Adjacent sequences: A362853 A362854 A362855 * A362857 A362858 A362859

Keyword

sign,tabl

Author

Seiichi Manyama, May 05 2023

Status

approved