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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. 15
 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, -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, 0, 1, -9, 20, -7, 42, -76, 14, -54, 38, -22, 0, 1, -10, 27, -16, 49, -132, 35, -104, 84, -98, 12, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..200, flattened FORMULA G.f. of column k: Product_{j>=1} 1/(1 + j*x^j)^k. EXAMPLE 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 + ... Square array begins:   1,  1,  1,   1,   1,   1,  ...   0, -1, -2,  -3,  -4,  -5,  ...   0, -1, -1,   0,   2,   5,  ...   0, -2, -2,  -1,   0,   0,  ...   0,  2,  9,  18,  27,  35,  ...   0, -1, -2, -12, -36, -76,  ... MAPLE with(numtheory): A:= proc(n, k) option remember; `if`(n=0, 1, -k*add(add(       (-d)^(1+j/d), d=divisors(j))*A(n-j, k), j=1..n)/n)     end: seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Apr 20 2018 MATHEMATICA 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 CROSSREFS 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. Main diagonal gives A297326. Antidiagonal sums give A299210. Cf. A266971, A297321, A297323, A297328. Sequence in context: A047265 A185962 A279928 * A278528 A257261 A355141 Adjacent sequences:  A297322 A297323 A297324 * A297326 A297327 A297328 KEYWORD sign,tabl AUTHOR Ilya Gutkovskiy, Dec 28 2017 STATUS approved

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Last modified August 15 02:47 EDT 2022. Contains 356122 sequences. (Running on oeis4.)