OFFSET
0,9
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
T(n,k) = Sum_{j=1..n} k^(n-j) * (n!/j!) * binomial(n-1,j-1) for n > 0.
T(n,k) = (2*k*n-2*k+1) * T(n-1,k) - k^2 * (n-1) * (n-2) * T(n-2,k) for n > 1.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, ...
1, 13, 37, 73, 121, 181, ...
1, 73, 361, 1009, 2161, 3961, ...
1, 501, 4361, 17341, 48081, 108101, ...
MATHEMATICA
T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 03 2021 *)
PROG
(PARI) {T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}
(PARI) {T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Feb 03 2021
STATUS
approved