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A351776 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!. 5
1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
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, 0, 4, 12, 24, 40, ...
0, 3, -6, -63, -204, -465, ...
0, -4, -8, 420, 2288, 7180, ...
0, -25, 150, -3435, -32020, -138525, ...
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, A302397, A351777, A351778.
Main diagonal gives A351779.
Sequence in context: A242464 A273185 A373183 * A259784 A145224 A138157
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Feb 19 2022
STATUS
approved

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Last modified May 30 04:46 EDT 2024. Contains 372958 sequences. (Running on oeis4.)