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A351790
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))^j/j!.
4
1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 6, 21, 24, 1, 1, 8, 42, 148, 120, 1, 1, 10, 69, 392, 1305, 720, 1, 1, 12, 102, 780, 4600, 13806, 5040, 1, 1, 14, 141, 1336, 11145, 64752, 170401, 40320, 1, 1, 16, 186, 2084, 22200, 191178, 1063216, 2403640, 362880
OFFSET
0,6
FORMULA
E.g.f. of column k: 1/(1 - x*exp(k*x)).
T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} k^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 4, 6, 8, 10, 12, ...
6, 21, 42, 69, 102, 141, ...
24, 148, 392, 780, 1336, 2084, ...
120, 1305, 4600, 11145, 22200, 39145, ...
MATHEMATICA
T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
PROG
(PARI) T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
(PARI) T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));
CROSSREFS
Columns k=0..4 give A000142, A006153, A336950, A336951, A336952.
Main diagonal gives A235328.
Sequence in context: A295259 A255009 A156579 * A322266 A190284 A327639
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Feb 19 2022
STATUS
approved