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A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)*log(1 + x))/(1 - k*log(1 + x)).

%I A334369 #33 May 01 2021 17:39:45
%S A334369 1,1,1,1,1,-1,1,1,1,2,1,1,3,2,-6,1,1,5,14,4,24,1,1,7,38,86,14,-120,1,
%T A334369 1,9,74,384,664,38,720,1,1,11,122,1042,4854,6136,216,-5040,1,1,13,182,
%U A334369 2204,18344,73614,66240,600,40320,1,1,15,254,4014,49774,387512,1302552,816672,6240,-362880
%N A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)*log(1 + x))/(1 - k*log(1 + x)).
%F A334369 T(0,k)=1 and T(n,k) = Sum_{j=0..n} j! * k^(j-1) * Stirling1(n,j) for n > 0.
%e A334369 Square array begins:
%e A334369    1,  1,   1,    1,     1,     1, ...
%e A334369    1,  1,   1,    1,     1,     1, ...
%e A334369   -1,  1,   3,    5,     7,     9, ...
%e A334369    2,  2,  14,   38,    74,   122, ...
%e A334369   -6,  4,  86,  384,  1042,  2204, ...
%e A334369   24, 14, 664, 4854, 18344, 49774, ...
%t A334369 T[0, k_] = 1; T[n_, k_] := Sum[If[k == 0 && j <= 1, 1, k^(j - 1)] * j! * StirlingS1[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 01 2021 *)
%Y A334369 Columns k=1..3 give A006252, A308878, A335530.
%Y A334369 Main diagonal gives A335529.
%Y A334369 Cf. A320080.
%K A334369 sign,tabl
%O A334369 0,10
%A A334369 _Seiichi Manyama_, Jun 12 2020