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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)|.
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%I #16 Oct 18 2022 13:31:56

%S 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 36,340,6578,0,1,0,0,0,24,210,3308,72792,0,1,0,0,0,0,240,2070,36288,

%U 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 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 For k > 0, e.g.f. of column k: 1/(1 - (-log(1-x))^k).

%F T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * |Stirling1(j,k)| * T(n-j,k).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 0, 0, 0, 0, ...

%e 0, 3, 2, 0, 0, 0, ...

%e 0, 14, 6, 6, 0, 0, ...

%e 0, 88, 46, 36, 24, 0, ...

%e 0, 694, 340, 210, 240, 120, ...

%o (PARI) T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1)));

%o (PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(-log(1-x+x*O(x^n)))^k), n));

%Y Columns k=0-5 give: A000007, A007840, A052811, A353118, A353119, A353200.

%Y Cf. A357119, A357868, A357882.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, Oct 18 2022