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A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|.

%I A357881 #16 Oct 18 2022 13:31:56
%S A357881 1,1,0,1,1,0,1,0,3,0,1,0,2,14,0,1,0,0,6,88,0,1,0,0,6,46,694,0,1,0,0,0,
%T A357881 36,340,6578,0,1,0,0,0,24,210,3308,72792,0,1,0,0,0,0,240,2070,36288,
%U A357881 920904,0,1,0,0,0,0,120,2040,24864,460752,13109088,0,1,0,0,0,0,0,1800,17640,310632,6551424,207360912,0
%N A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|.
%F A357881 For k > 0, e.g.f. of column k: 1/(1 - (-log(1-x))^k).
%F A357881 T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * |Stirling1(j,k)| * T(n-j,k).
%e A357881 Square array begins:
%e A357881   1,   1,   1,   1,   1,   1, ...
%e A357881   0,   1,   0,   0,   0,   0, ...
%e A357881   0,   3,   2,   0,   0,   0, ...
%e A357881   0,  14,   6,   6,   0,   0, ...
%e A357881   0,  88,  46,  36,  24,   0, ...
%e A357881   0, 694, 340, 210, 240, 120, ...
%o A357881 (PARI) T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1)));
%o A357881 (PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(-log(1-x+x*O(x^n)))^k), n));
%Y A357881 Columns k=0-5 give: A000007, A007840, A052811, A353118, A353119, A353200.
%Y A357881 Cf. A357119, A357868, A357882.
%K A357881 nonn,tabl
%O A357881 0,9
%A A357881 _Seiichi Manyama_, Oct 18 2022