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 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. 14
 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, -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, 12, 42, -21, -26, -90, -33, -31, 0, 0, 1, -10, 18, 48, -56, -66, -145, -56, -66, -72, 2, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS FORMULA G.f. of column k: Product_{j>=1} (1 - j*x^j)^k. EXAMPLE 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 + ... Square array begins:   1,  1,   1,   1,   1,   1,  ...   0, -1,  -2,  -3,  -4,  -5,  ...   0, -2,  -3,  -3,  -2,   0,  ...   0, -1,   2,   8,  16,  25,  ...   0, -1,   4,   9,   9,   0,  ...   0,  5,  16,  18,   4, -26,  ... MATHEMATICA 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 PROG (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 CROSSREFS 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. Main diagonal gives A297324. Antidiagonal sums give A299209. Cf. A266964, A297321, A297325, A297328. Sequence in context: A104245 A185287 A276554 * A257654 A167637 A109754 Adjacent sequences:  A297320 A297321 A297322 * A297324 A297325 A297326 KEYWORD sign,tabl AUTHOR Ilya Gutkovskiy, Dec 28 2017 STATUS approved

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Last modified January 18 13:05 EST 2019. Contains 319271 sequences. (Running on oeis4.)