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A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

%I A294042 #35 Jan 31 2024 08:05:54
%S A294042 1,1,0,1,1,0,1,2,1,0,1,3,6,1,0,1,4,15,20,1,0,1,5,28,87,76,1,0,1,6,45,
%T A294042 232,585,312,1,0,1,7,66,485,2248,4383,1384,1,0,1,8,91,876,6145,24544,
%U A294042 35919,6512,1,0,1,9,120,1435,13716,88245,295456,318195,32400,1,0
%N A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).
%H A294042 Seiichi Manyama, <a href="/A294042/b294042.txt">Antidiagonals n = 0..139, flattened</a>
%F A294042 A(0,k) = 1 and A(n,k) = k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * A(n-j,k)/(n-j)! for n > 0.
%F A294042 A(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * Bell(j). - _Seiichi Manyama_, Jan 31 2024
%e A294042 Square array A(n,k) begins:
%e A294042    1, 1,   1,    1,     1,     1, ...
%e A294042    0, 1,   2,    3,     4,     5, ...
%e A294042    0, 1,   6,   15,    28,    45, ...
%e A294042    0, 1,  20,   87,   232,   485, ...
%e A294042    0, 1,  76,  585,  2248,  6145, ...
%e A294042    0, 1, 312, 4383, 24544, 88245, ...
%Y A294042 Columns k=0..5 give A000007, A000012, A000898, A192989, A202824, A202825.
%Y A294042 Rows n=0..2 give A000012, A001477, A000384.
%Y A294042 Main diagonal gives A294045.
%Y A294042 Cf. A000110, A293991.
%K A294042 nonn,tabl
%O A294042 0,8
%A A294042 _Seiichi Manyama_, Oct 22 2017