A297325 - OEIS

%I #16 Apr 20 2018 09:37:48

%S 1,1,0,1,-1,0,1,-2,-1,0,1,-3,-1,-2,0,1,-4,0,-2,2,0,1,-5,2,-1,9,-1,0,1,

%T -6,5,0,18,-2,4,0,1,-7,9,0,27,-12,10,-1,0,1,-8,14,-2,35,-36,11,-16,18,

%U 0,1,-9,20,-7,42,-76,14,-54,38,-22,0,1,-10,27,-16,49,-132,35,-104,84,-98,12,0

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

%H Alois P. Heinz, <a href="/A297325/b297325.txt">Antidiagonals n = 0..200, flattened</a>

%F G.f. of column k: Product_{j>=1} 1/(1 + j*x^j)^k.

%e G.f. of column k: A_k(x) = 1 - k*x + (1/2)*k*(k - 3)*x^2 - (1/6)*k*(k^2 - 9*k + 20)*x^3 + (1/24)*k*(k^3 - 18*k^2 + 107*k - 42)*x^4 - (1/120)*k*(k^4 - 30*k^3 + 335*k^2 - 810*k + 624)*x^5 + ...

%e Square array begins:

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

%e   0, -1, -2,  -3,  -4,  -5,  ...

%e   0, -1, -1,   0,   2,   5,  ...

%e   0, -2, -2,  -1,   0,   0,  ...

%e   0,  2,  9,  18,  27,  35,  ...

%e   0, -1, -2, -12, -36, -76,  ...

%p with(numtheory):

%p A:= proc(n, k) option remember; `if`(n=0, 1, -k*add(add(

%p       (-d)^(1+j/d), d=divisors(j))*A(n-j, k), j=1..n)/n)

%p     end:

%p seq(seq(A(n, d-n), n=0..d), d=0..14);  # _Alois P. Heinz_, Apr 20 2018

%t Table[Function[k, SeriesCoefficient[Product[1/(1 + i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

%Y Columns k=0..32 give A000007, A022693, A022694, A022695, A022696, A022697, A022698, A022699, A022700, A022701, A022702, A022703, A022704, A022705, A022706, A022707, A022708, A022709, A022710, A022711, A022712, A022713, A022714, A022715, A022716, A022717, A022718, A022719, A022720, A022721, A022722, A022723, A022724.

%Y Main diagonal gives A297326.

%Y Antidiagonal sums give A299210.

%Y Cf. A266971, A297321, A297323, A297328.

%K sign,tabl

%O 0,8

%A _Ilya Gutkovskiy_, Dec 28 2017