cp's OEIS Frontend

A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.

%I A297323 #15 Feb 06 2018 11:45:21
%S A297323 1,1,0,1,-1,0,1,-2,-2,0,1,-3,-3,-1,0,1,-4,-3,2,-1,0,1,-5,-2,8,4,5,0,1,
%T A297323 -6,0,16,9,16,1,0,1,-7,3,25,9,18,-3,13,0,1,-8,7,34,0,4,-35,6,4,0,1,-9,
%U A297323 12,42,-21,-26,-90,-33,-31,0,0,1,-10,18,48,-56,-66,-145,-56,-66,-72,2,0
%N A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.
%F A297323 G.f. of column k: Product_{j>=1} (1 - j*x^j)^k.
%e A297323 G.f. of column k: A_k(x) = 1 - k*x + (1/2)*k*(k - 5)*x^2 - (1/6)*k*(k^2 - 15*k + 20)*x^3 + (1/24)*k*(k^3 - 30*k^2 + 155*k - 150)*x^4 - (1/120)*k*(k^4 - 50*k^3 + 575*k^2 - 1750*k + 624)*x^5 + ...
%e A297323 Square array begins:
%e A297323   1,  1,   1,   1,   1,   1,  ...
%e A297323   0, -1,  -2,  -3,  -4,  -5,  ...
%e A297323   0, -2,  -3,  -3,  -2,   0,  ...
%e A297323   0, -1,   2,   8,  16,  25,  ...
%e A297323   0, -1,   4,   9,   9,   0,  ...
%e A297323   0,  5,  16,  18,   4, -26,  ...
%t A297323 Table[Function[k, SeriesCoefficient[Product[(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%o A297323 (PARI) first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, (1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ _Iain Fox_, Dec 28 2017
%Y A297323 Columns k=0..32 give A000007, A022661, A022662, A022663, A022664, A022665, A022666, A022667, A022668, A022669, A022670, A022671, A022672, A022673, A022674, A022675, A022676, A022677, A022678, A022679, A022680, A022681, A022682, A022683, A022684, A022685, A022686, A022687, A022688, A022689, A022690, A022691, A022692.
%Y A297323 Main diagonal gives A297324.
%Y A297323 Antidiagonal sums give A299209.
%Y A297323 Cf. A266964, A297321, A297325, A297328.
%K A297323 sign,tabl
%O A297323 0,8
%A A297323 _Ilya Gutkovskiy_, Dec 28 2017