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A351420 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - f^(k-1)(x)), where f(x) = log(1+x).

%I A351420 #22 May 12 2023 12:40:53
%S A351420 1,1,1,1,0,2,1,-1,1,6,1,-2,3,-1,24,1,-3,8,-13,8,120,1,-4,16,-48,77,
%T A351420 -26,720,1,-5,27,-124,386,-576,194,5040,1,-6,41,-259,1270,-3905,5219,
%U A351420 -1142,40320,1,-7,58,-471,3244,-16243,47701,-55567,9736,362880
%N A351420 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - f^(k-1)(x)), where f(x) = log(1+x).
%F A351420 T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.
%e A351420 Square array begins:
%e A351420     1,   1,    1,     1,      1,      1, ...
%e A351420     1,   0,   -1,    -2,     -3,     -4, ...
%e A351420     2,   1,    3,     8,     16,     27, ...
%e A351420     6,  -1,  -13,   -48,   -124,   -259, ...
%e A351420    24,   8,   77,   386,   1270,   3244, ...
%e A351420   120, -26, -576, -3905, -16243, -50375, ...
%t A351420 T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Feb 11 2022 *)
%o A351420 (PARI) T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));
%Y A351420 Columns k=1..5 give A000142(n-1), (-1)^(n-1) * A089064(n), A351421, A351422, A351423.
%Y A351420 Main diagonal gives A351424.
%Y A351420 Cf. A111933, A351429.
%K A351420 sign,tabl
%O A351420 1,6
%A A351420 _Seiichi Manyama_, Feb 11 2022