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A383341 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k,j)/(n-j)!.

%I A383341 #21 Apr 25 2025 16:00:41
%S A383341 1,1,1,1,1,2,1,1,3,6,1,1,4,11,24,1,1,5,16,53,120,1,1,6,21,88,309,720,
%T A383341 1,1,7,26,129,568,2119,5040,1,1,8,31,176,897,4288,16687,40320,1,1,9,
%U A383341 36,229,1296,7317,36832,148329,362880,1,1,10,41,288,1765,11296,67365,354688,1468457,3628800
%N A383341 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k,j)/(n-j)!.
%F A383341 E.g.f. of column k: exp(-k*x) / (1-x)^(k+1).
%F A383341 A(0,k) = A(1,k) = 1; A(n,k) = n*A(n-1,k) + k*(n-1)*A(n-2,k).
%e A383341 Square array begins:
%e A383341     1,    1,    1,    1,     1,     1,     1, ...
%e A383341     1,    1,    1,    1,     1,     1,     1, ...
%e A383341     2,    3,    4,    5,     6,     7,     8, ...
%e A383341     6,   11,   16,   21,    26,    31,    36, ...
%e A383341    24,   53,   88,  129,   176,   229,   288, ...
%e A383341   120,  309,  568,  897,  1296,  1765,  2304, ...
%e A383341   720, 2119, 4288, 7317, 11296, 16315, 22464, ...
%o A383341 (PARI) a(n,k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k, j)/(n-j)!);
%Y A383341 Columns k=0..4 give A000142, A000255, A052124, A383378, A383383.
%Y A383341 Main diagonal gives A383379.
%Y A383341 Cf. A295181.
%K A383341 nonn,tabl
%O A383341 0,6
%A A383341 _Seiichi Manyama_, Apr 24 2025