OFFSET
0,8
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, k). - Peter Luschny, Dec 23 2021
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 6, 12, 20, 30, 42, ...
0, 5, 22, 57, 116, 205, 330, ...
0, 15, 94, 309, 756, 1555, 2850, ...
0, 52, 454, 1866, 5428, 12880, 26682, ...
0, 203, 2430, 12351, 42356, 115155, 268098, ...
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
end:
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 25 2017
MATHEMATICA
A[0, _] = 1; A[n_ /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[_, _] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Sep 25 2017
STATUS
approved