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A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

%I A355607 #21 Jul 11 2022 03:36:13
%S A355607 1,1,1,1,0,0,1,0,2,0,1,0,0,-3,0,1,0,0,6,20,0,1,0,0,0,-12,-90,0,1,0,0,
%T A355607 0,24,40,594,0,1,0,0,0,0,-60,180,-4200,0,1,0,0,0,0,120,240,-1512,
%U A355607 34544,0,1,0,0,0,0,0,-360,-1260,11760,-316008,0,1,0,0,0,0,0,720,1680,28224,-38880,3207240,0
%N A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).
%H A355607 Seiichi Manyama, <a href="/A355607/b355607.txt">Antidiagonals n = 0..139, flattened</a>
%F A355607 T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
%F A355607 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(n-k*j)!.
%e A355607 Square array begins:
%e A355607   1,   1,   1,   1,    1,   1, 1, ...
%e A355607   1,   0,   0,   0,    0,   0, 0, ...
%e A355607   0,   2,   0,   0,    0,   0, 0, ...
%e A355607   0,  -3,   6,   0,    0,   0, 0, ...
%e A355607   0,  20, -12,  24,    0,   0, 0, ...
%e A355607   0, -90,  40, -60,  120,   0, 0, ...
%e A355607   0, 594, 180, 240, -360, 720, 0, ...
%o A355607 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(n-k*j)!);
%Y A355607 Columns k=1..4 give A007113, A007121, (-1)^n * A353229(n), A354625.
%Y A355607 Cf. A292892, A355609, A355619.
%K A355607 sign,tabl,look
%O A355607 0,9
%A A355607 _Seiichi Manyama_, Jul 09 2022