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A292860 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)).

%I A292860 #53 Dec 23 2021 06:05:32
%S A292860 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 A292860 30,116,309,454,203,0,1,7,42,205,756,1866,2430,877,0,1,8,56,330,1555,
%U A292860 5428,12351,14214,4140,0,1,9,72,497,2850,12880,42356,88563,89918,21147,0
%N A292860 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 A292860 Seiichi Manyama, <a href="/A292860/b292860.txt">Antidiagonals n = 0..139, flattened</a>
%F A292860 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 A292860 A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - _Seiichi Manyama_, Jul 27 2019
%F A292860 A(n,k) = BellPolynomial(n, k). - _Peter Luschny_, Dec 23 2021
%e A292860 Square array begins:
%e A292860    1,   1,    1,     1,     1,      1,      1, ...
%e A292860    0,   1,    2,     3,     4,      5,      6, ...
%e A292860    0,   2,    6,    12,    20,     30,     42, ...
%e A292860    0,   5,   22,    57,   116,    205,    330, ...
%e A292860    0,  15,   94,   309,   756,   1555,   2850, ...
%e A292860    0,  52,  454,  1866,  5428,  12880,  26682, ...
%e A292860    0, 203, 2430, 12351, 42356, 115155, 268098, ...
%p A292860 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A292860       (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
%p A292860     end:
%p A292860 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Sep 25 2017
%t A292860 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 A292860 Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 13 2021 *)
%t A292860 A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Peter Luschny_, Dec 23 2021 *)
%Y A292860 Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.
%Y A292860 Rows n=0..2 give A000012, A001477, A002378.
%Y A292860 Main diagonal gives A242817.
%Y A292860 Same array, different indexing is A189233.
%Y A292860 Cf. A292861.
%K A292860 nonn,tabl
%O A292860 0,8
%A A292860 _Seiichi Manyama_, Sep 25 2017