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A308484 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} j^k * x^j/j!).

%I A308484 #22 May 12 2021 03:56:29
%S A308484 1,1,0,1,1,0,1,3,-1,0,1,7,-1,-2,0,1,15,5,-26,9,0,1,31,35,-146,29,6,0,
%T A308484 1,63,149,-650,-351,756,-155,0,1,127,539,-2642,-5251,9936,-1793,232,0,
%U A308484 1,255,1805,-10346,-46071,83376,51421,-45744,3969,0
%N A308484 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} j^k * x^j/j!).
%H A308484 Seiichi Manyama, <a href="/A308484/b308484.txt">Antidiagonals n = 1..140, flattened</a>
%F A308484 A(n,k) = n^k - Sum_{j=1..n-1} binomial(n-1,j)*j^k*A(n-j,k).
%e A308484 Square array begins:
%e A308484    1,    1,     1,     1,       1,        1, ...
%e A308484    0,    1,     3,     7,      15,       31, ...
%e A308484    0,   -1,    -1,     5,      35,      149, ...
%e A308484    0,   -2,   -26,  -146,    -650,    -2642, ...
%e A308484    0,    9,    29,  -351,   -5251,   -46071, ...
%e A308484    0,    6,   756,  9936,   83376,   559656, ...
%e A308484    0, -155, -1793, 51421, 1623439, 28735405, ...
%t A308484 T[n_, k_] := T[n, k] = n^k - Sum[Binomial[n-1,j] * j^k * T[n-j,k], {j,1,n-1}]; Table[T[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 12 2021 *)
%Y A308484 Columns k=0..4 give A000007(n-1), A009306, A033464, A300452, A306325.
%Y A308484 A(n,n) gives A320939.
%K A308484 sign,tabl
%O A308484 1,8
%A A308484 _Seiichi Manyama_, May 30 2019