login
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).
2

%I #20 Nov 10 2017 22:10:22

%S 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 -72,180,-131,1,-1,-32,-114,-360,840,-1080,1499,1,-1,-64,-390,-1656,

%U 4200,-8640,15120,-4159,1,-1,-128,-1266,-7272,22440,-69120,161280,-45360

%N 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 Seiichi Manyama, <a href="/A294616/b294616.txt">Antidiagonals n = 0..139, flattened</a>

%F 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 Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e -1, -1, -1, -1, -1, -1, ...

%e -1, -2, -4, -8, -16, -32, ...

%e 1, 0, -6, -30, -114, -390, ...

%e -1, -12, -72, -360, -1656, -7272, ...

%e 41, 180, 840, 4200, 22440, 126600, ...

%Y Columns k=0..1 give A028343, A294463.

%Y Rows n=0..3 give A000012, (-1)*A000012, (-1)*A000079, (-1)*A245804.

%Y Cf. A294761.

%K sign,tabl

%O 0,9

%A _Seiichi Manyama_, Nov 05 2017