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 A297321 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. 28
 1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 7, 0, 1, 5, 14, 28, 28, 15, 0, 1, 6, 20, 48, 69, 64, 25, 0, 1, 7, 27, 75, 137, 174, 133, 43, 0, 1, 8, 35, 110, 240, 380, 413, 266, 64, 0, 1, 9, 44, 154, 387, 726, 998, 933, 513, 120, 0, 1, 10, 54, 208, 588, 1266, 2075, 2488, 2046, 1000, 186, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened 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 + 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,   2,   5,    9,   14,   20,  ... 0,   5,  14,   28,   48,   75,  ... 0,   7,  28,   69,  137,  240,  ... 0,  15,  64,  174,  380,  726,  ... 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 CROSSREFS Columns k=0..32 give A000007, A022629, A022630, A022631, A022632, A022633, A022634, A022635, A022636, A022637, A022638, A022639, A022640, A022641, A022642, A022643, A022644, A022645, A022646, A022647, A022648, A022649, A022650, A022651, A022652, A022653, A022654, A022655, A022656, A022657, A022658, A022659, A022660. Main diagonal gives A297322. Antidiagonal sums give A299164. Cf. A266891, A297323, A297325, A297328. Sequence in context: A144064 A172236 A191646 * A277938 A130020 A292870 Adjacent sequences:  A297318 A297319 A297320 * A297322 A297323 A297324 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, Dec 28 2017 STATUS approved

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Last modified November 30 20:17 EST 2021. Contains 349425 sequences. (Running on oeis4.)