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A355619 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/k!).

%I A355619 #17 Jul 11 2022 03:36:09
%S A355619 1,1,1,1,0,0,1,0,2,0,1,0,0,-3,0,1,0,0,3,20,0,1,0,0,0,-6,-90,0,1,0,0,0,
%T A355619 4,20,594,0,1,0,0,0,0,-10,0,-4200,0,1,0,0,0,0,5,40,-126,34544,0,1,0,0,
%U A355619 0,0,0,-15,-210,1260,-316008,0,1,0,0,0,0,0,6,70,1904,-4320,3207240,0
%N A355619 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/k!).
%H A355619 Seiichi Manyama, <a href="/A355619/b355619.txt">Antidiagonals n = 0..139, flattened</a>
%F A355619 T(0,k) = 1 and T(n,k) = -(n-1)!/k! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
%F A355619 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(k!^j * (n-k*j)!).
%e A355619 Square array begins:
%e A355619   1,   1,  1,   1,   1, 1, 1, ...
%e A355619   1,   0,  0,   0,   0, 0, 0, ...
%e A355619   0,   2,  0,   0,   0, 0, 0, ...
%e A355619   0,  -3,  3,   0,   0, 0, 0, ...
%e A355619   0,  20, -6,   4,   0, 0, 0, ...
%e A355619   0, -90, 20, -10,   5, 0, 0, ...
%e A355619   0, 594,  0,  40, -15, 6, 0, ...
%o A355619 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(k!^j*(n-k*j)!));
%Y A355619 Columns k=1..4 give A007113, A355605, (-1)^n * A351493(n), A355603.
%Y A355619 Cf. A355607, A355610.
%K A355619 sign,tabl
%O A355619 0,9
%A A355619 _Seiichi Manyama_, Jul 10 2022