A294808 - OEIS

A294808

Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j)^j in powers of x.

%I #21 Nov 18 2017 04:21:39

%S 1,1,-1,1,-1,-2,1,-1,-8,-1,1,-1,-32,-73,0,1,-1,-128,-2155,-927,4,1,-1,

%T -512,-58921,-259701,-13969,4,1,-1,-2048,-1593811,-67045719,-48496253,

%U -254580,7,1,-1,-8192,-43044673,-17178209325,-152513227585,-13001952944,-5288596,3

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j)^j in powers of x.

%H Seiichi Manyama, <a href="/A294808/b294808.txt">Antidiagonals n = 0..52, flattened</a>

%F A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j)) * A(n-j,k) for n > 0.

%e Square array begins:

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

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

%e -2, -8, -32, -128, -512, ...

%e -1, -73, -2155, -58921, -1593811, ...

%e 0, -927, -259701, -67045719, -17178209325, ...

%e 4, -13969, -48496253, -152513227585, -476819162106101, ...

%Y Columns k=0..2 give A073592, A294809, A294953.

%Y Rows n=0..2 give A000012, (-1)*A000012, (-1)*A004171.

%Y Cf. A294653, A294950.

%K sign,tabl

%O 0,6

%A _Seiichi Manyama_, Nov 09 2017