%I #53 Dec 23 2021 06:05:32

%S 1,1,0,1,1,0,1,2,2,0,1,3,6,5,0,1,4,12,22,15,0,1,5,20,57,94,52,0,1,6,

%T 30,116,309,454,203,0,1,7,42,205,756,1866,2430,877,0,1,8,56,330,1555,

%U 5428,12351,14214,4140,0,1,9,72,497,2850,12880,42356,88563,89918,21147,0

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

%H Seiichi Manyama, <a href="/A292860/b292860.txt">Antidiagonals n = 0..139, flattened</a>

%F A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.

%F A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - _Seiichi Manyama_, Jul 27 2019

%F A(n,k) = BellPolynomial(n, k). - _Peter Luschny_, Dec 23 2021

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 3, 4, 5, 6, ...

%e 0, 2, 6, 12, 20, 30, 42, ...

%e 0, 5, 22, 57, 116, 205, 330, ...

%e 0, 15, 94, 309, 756, 1555, 2850, ...

%e 0, 52, 454, 1866, 5428, 12880, 26682, ...

%e 0, 203, 2430, 12351, 42356, 115155, 268098, ...

%p A:= proc(n, k) option remember; `if`(n=0, 1,

%p (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)

%p end:

%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Sep 25 2017

%t 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;

%t Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 13 2021 *)

%t A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Peter Luschny_, Dec 23 2021 *)

%Y Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.

%Y Rows n=0..2 give A000012, A001477, A002378.

%Y Main diagonal gives A242817.

%Y Same array, different indexing is A189233.

%Y Cf. A292861.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Sep 25 2017