login
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).
3
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
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
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, May 05 2023
STATUS
approved