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 A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k. 35
 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part. LINKS Seiichi Manyama, Antidiagonals n = 0..139, flattened N. J. A. Sloane, Transforms FORMULA G.f. of column k: Product_{j>=1} (1 + x^j)^k. A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019 EXAMPLE A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part). Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part). Square array begins:   1,  1,  1,   1,   1,   1,  ...   0,  1,  2,   3,   4,   5,  ...   0,  1,  3,   6,  10,  15,  ...   0,  2,  6,  13,  24,  40,  ...   0,  2,  9,  24,  51,  95,  ...   0,  3, 14,  42, 100, 206,  ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(      (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))     end: A:= (n, k)-> b(n\$2, k): seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019 MATHEMATICA Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten CROSSREFS Columns k=0-32 give: A000007, A000009, A022567-A022596. Rows n=0-2 give: A000012, A001477, A000217. Main diagonal gives A270913. Antidiagonal sums give A299106. Cf. A144064, A286352, A308680. Sequence in context: A017827 A279778 A094266 * A291652 A071569 A261835 Adjacent sequences:  A286332 A286333 A286334 * A286336 A286337 A286338 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, May 07 2017 STATUS approved

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Last modified September 22 06:53 EDT 2020. Contains 337289 sequences. (Running on oeis4.)