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%I #27 Nov 03 2017 11:53:33
%S 1,1,1,1,1,1,1,1,2,1,1,1,3,4,1,1,1,4,9,10,1,1,1,5,16,33,26,1,1,1,6,25,
%T 76,141,76,1,1,1,7,36,145,436,651,232,1,1,1,8,49,246,1025,2776,3333,
%U 764,1,1,1,9,64,385,2046,8245,19384,18369,2620,1,1,1,10,81
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j=1..k+1} binomial(k,j-1)*x^j/j).
%H Seiichi Manyama, <a href="/A293991/b293991.txt">Antidiagonals n = 0..139, flattened</a>
%F E.g.f. of column k: exp(((1+x)^(k+1) - 1)/(k+1)).
%F A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k+1,n)} binomial(k,j-1)*A(n-j,k)/(n-j)! for n > 0.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, ...
%e 1, 4, 9, 16, 25, ...
%e 1, 10, 33, 76, 145, ...
%e 1, 26, 141, 436, 1025, ...
%t A[0, _] = 1; A[n_, k_] := (n-1)!*Sum[Binomial[k, j-1]*A[n-j, k]/(n-j)!, {j, 1, Min[k+1, n]}];
%t Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 03 2017 *)
%Y Columns k=0..5 give A000012, A000085, A049425, A049426, A049427, A049428.
%Y Rows n=0-1 give A000012.
%Y Main diagonal gives A294003.
%Y Cf. A291709.
%K nonn,tabl
%O 0,9
%A _Seiichi Manyama_, Oct 21 2017
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