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A294616 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: Product_{j>0} (1-j^k*x^j)^(1/j).

%I A294616 #20 Nov 10 2017 22:10:22
%S A294616 1,1,-1,1,-1,-1,1,-1,-2,1,1,-1,-4,0,-1,1,-1,-8,-6,-12,41,1,-1,-16,-30,
%T A294616 -72,180,-131,1,-1,-32,-114,-360,840,-1080,1499,1,-1,-64,-390,-1656,
%U A294616 4200,-8640,15120,-4159,1,-1,-128,-1266,-7272,22440,-69120,161280,-45360
%N A294616 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: Product_{j>0} (1-j^k*x^j)^(1/j).
%H A294616 Seiichi Manyama, <a href="/A294616/b294616.txt">Antidiagonals n = 0..139, flattened</a>
%F A294616 A(0,k) = 1 and A(n,k) = -(n-1)! * Sum_{j=1..n} (Sum_{d|j} d^(k*j/d)) * A(n-j,k)/(n-j)! for n > 0.
%e A294616 Square array begins:
%e A294616     1,   1,   1,    1,     1,      1, ...
%e A294616    -1,  -1,  -1,   -1,    -1,     -1, ...
%e A294616    -1,  -2,  -4,   -8,   -16,    -32, ...
%e A294616     1,   0,  -6,  -30,  -114,   -390, ...
%e A294616    -1, -12, -72, -360, -1656,  -7272, ...
%e A294616    41, 180, 840, 4200, 22440, 126600, ...
%Y A294616 Columns k=0..1 give A028343, A294463.
%Y A294616 Rows n=0..3 give A000012, (-1)*A000012, (-1)*A000079, (-1)*A245804.
%Y A294616 Cf. A294761.
%K A294616 sign,tabl
%O A294616 0,9
%A A294616 _Seiichi Manyama_, Nov 05 2017