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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k/k! * log(1 - x)).
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%I #41 Jul 13 2022 12:07:25

%S 1,1,1,1,0,3,1,0,2,14,1,0,0,3,88,1,0,0,3,32,694,1,0,0,0,6,150,6578,1,

%T 0,0,0,4,20,1524,72792,1,0,0,0,0,10,270,12600,920904,1,0,0,0,0,5,40,

%U 1764,147328,13109088,1,0,0,0,0,0,15,210,12600,1705536,207360912,1,0,0,0,0,0,6,70,2464,146880,23681520,3608233056

%N Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k/k! * log(1 - x)).

%H Amiram Eldar, <a href="/A355652/b355652.txt">Antidiagonals n = 0..150, flattened</a>

%F T(0,k) = 1 and T(n,k) = (n!/k!) * Sum_{j=k+1..n} 1/(j-k) * T(n-j,k)/(n-j)! for n > 0.

%F T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(k!^j * (n-k*j)!).

%e Square array begins:

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

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

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

%e 14, 3, 3, 0, 0, 0, 0, ...

%e 88, 32, 6, 4, 0, 0, 0, ...

%e 694, 150, 20, 10, 5, 0, 0, ...

%e 6578, 1524, 270, 40, 15, 6, 0, ...

%t T[n_, k_] := n! * Sum[j! * Abs[StirlingS1[n - k*j, j]]/(k!^j*(n - k*j)!), {j, 0, Floor[n/(k + 1)]}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jul 13 2022 *)

%o (PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(k!^j*(n-k*j)!));

%Y Columns k=0..3 give A007840, A052830, A351505, A351506.

%Y Cf. A355610, A355665.

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Jul 13 2022