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A297325 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.

%I A297325 #16 Apr 20 2018 09:37:48
%S A297325 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 A297325 -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 A297325 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 A297325 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 A297325 Alois P. Heinz, <a href="/A297325/b297325.txt">Antidiagonals n = 0..200, flattened</a>
%F A297325 G.f. of column k: Product_{j>=1} 1/(1 + j*x^j)^k.
%e A297325 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 A297325 Square array begins:
%e A297325   1,  1,  1,   1,   1,   1,  ...
%e A297325   0, -1, -2,  -3,  -4,  -5,  ...
%e A297325   0, -1, -1,   0,   2,   5,  ...
%e A297325   0, -2, -2,  -1,   0,   0,  ...
%e A297325   0,  2,  9,  18,  27,  35,  ...
%e A297325   0, -1, -2, -12, -36, -76,  ...
%p A297325 with(numtheory):
%p A297325 A:= proc(n, k) option remember; `if`(n=0, 1, -k*add(add(
%p A297325       (-d)^(1+j/d), d=divisors(j))*A(n-j, k), j=1..n)/n)
%p A297325     end:
%p A297325 seq(seq(A(n, d-n), n=0..d), d=0..14);  # _Alois P. Heinz_, Apr 20 2018
%t A297325 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 A297325 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 A297325 Main diagonal gives A297326.
%Y A297325 Antidiagonal sums give A299210.
%Y A297325 Cf. A266971, A297321, A297323, A297328.
%K A297325 sign,tabl
%O A297325 0,8
%A A297325 _Ilya Gutkovskiy_, Dec 28 2017