login
A351761
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.
6
1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0
OFFSET
0,8
FORMULA
E.g.f. of column k: 1/(1 - k*x*exp(x)).
T(0,k) = 1 and T(n,k) = k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 4, 12, 24, 40, 60, ...
0, 21, 102, 279, 588, 1065, ...
0, 148, 1160, 4332, 11536, 25220, ...
0, 1305, 16490, 84075, 282900, 746525, ...
PROG
(PARI) T(n, k) = n!*sum(j=0, n, k^(n-j)*(n-j)^j/j!);
(PARI) T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
CROSSREFS
Columns k=0..3 give A000007, A006153, A351762, A351763.
Main diagonal gives A351765.
Sequence in context: A173004 A118343 A309148 * A226031 A308460 A244116
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Feb 18 2022
STATUS
approved