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A292894 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (1 - exp(x))).

%I A292894 #32 Jul 10 2022 08:09:13
%S A292894 1,1,-1,1,0,0,1,0,-2,1,1,0,0,-3,1,1,0,0,-6,8,-2,1,0,0,0,-12,55,-9,1,0,
%T A292894 0,0,-24,-20,84,-9,1,0,0,0,0,-60,330,-637,50,1,0,0,0,0,-120,-120,2478,
%U A292894 -4992,267,1,0,0,0,0,0,-360,-210,11704,-10593,413,1,0,0,0,0,0,-720,-840,19824,-15192,92060,-2180
%N A292894 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (1 - exp(x))).
%H A292894 Seiichi Manyama, <a href="/A292894/b292894.txt">Antidiagonals n = 0..139, flattened</a>
%F A292894 From _Seiichi Manyama_, Jul 09 2022: (Start)
%F A292894 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} (-1)^j * Stirling2(n-k*j,j)/(n-k*j)!.
%F A292894 T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)
%e A292894 Square array begins:
%e A292894    1,  1,   1,   1,    1, ...
%e A292894   -1,  0,   0,   0,    0, ...
%e A292894    0, -2,   0,   0,    0, ...
%e A292894    1, -3,  -6,   0,    0, ...
%e A292894    1,  8, -12, -24,    0, ...
%e A292894   -2, 55, -20, -60, -120, ...
%o A292894 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), (-1)^j*stirling(n-k*j, j, 2)/(n-k*j)!); \\ _Seiichi Manyama_, Jul 09 2022
%Y A292894 Columns k=0..2 give A000587, A292893, A292951.
%Y A292894 Rows n=0..1 give A000012, (-1)*A000007.
%Y A292894 Main diagonal gives A000007.
%Y A292894 Cf. A292892, A355607.
%K A292894 sign,tabl
%O A292894 0,9
%A A292894 _Seiichi Manyama_, Sep 26 2017