A293461 - OEIS

%I #21 Oct 10 2017 03:06:14

%S 1,1,0,1,1,0,1,1,0,0,1,1,2,1,0,1,1,2,1,1,0,1,1,2,4,1,1,0,1,1,2,4,1,3,

%T 1,0,1,1,2,4,5,3,3,1,0,1,1,2,4,5,3,6,5,2,0,1,1,2,4,5,8,6,5,6,2,0,1,1,

%U 2,4,5,8,6,9,9,4,2,0,1,1,2,4,5,8,12,9,9,13

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*(2*i-1))).

%H Seiichi Manyama, <a href="/A293461/b293461.txt">Antidiagonals n = 0..139, flattened</a>

%e Square array begins:

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

%e    0, 1, 1, 1, 1, ...

%e    0, 0, 2, 2, 2, ...

%e    0, 1, 1, 4, 4, ...

%e    0, 1, 1, 1, 5, ...

%e    0, 1, 3, 3, 3, ...

%t max = 12; A[n_, k_] := SeriesCoefficient[Product[(x*(-(k*x^((2*i - 1)*(k + 1) + 1)) - x^((2*i - 1)*(k + 1) + 1) + k*x^((2*i - 1)*(k + 1) + 2*i) + x^(2*i)))/(x^(2*i) - x)^2 + 1, {i, 1, max}], {x, 0, n}]; Flatten[ Table[ A[n - k, k], {n, 0, max}, {k, n, 0, -1}]] (* _Jean-François Alcover_, Oct 10 2017 *)

%Y Columns k=0..3 give A000007, A000700, A293304, A293463.

%Y Rows n=0..1 give A000012, A057427.

%Y Main diagonal gives A102186.

%Y Cf. A290216.

%K nonn,tabl

%O 0,13

%A _Seiichi Manyama_, Oct 09 2017