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 A286354 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. 6
 1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, 0, 0, 1, -4, 0, 2, 0, 0, 1, -5, 2, 5, 1, 1, 0, 1, -6, 5, 8, 0, 2, 0, 0, 1, -7, 9, 10, -5, 0, -2, 1, 0, 1, -8, 14, 10, -15, -4, -7, 0, 0, 0, 1, -9, 20, 7, -30, -6, -10, 0, -2, 0, 0, 1, -10, 27, 0, -49, 0, -5, 8, 0, -2, 0, 0, 1, -11, 35, -12, -70, 21, 11, 25, 9, 0, 1, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS A(n,k) number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts with k types of each part. LINKS Seiichi Manyama, Antidiagonals n = 0..139, flattened Index entries for expansions of Product_{k >= 1} (1-x^k)^m FORMULA G.f. of column k: Product_{j>=1} (1 - x^j)^k. G.f. of column k: (Sum_{j=-inf..inf} (-1)^j*x^(j*(3*j+1)/2))^k. Column k is the Euler transform of period 1 sequence [-k, -k, -k, ...]. EXAMPLE A(3,2) = 2 because we have [2, 1], [2', 1], [2, 1'], [2', 1'] (number of partitions of 3 into an even number of distinct parts with 2 types of each part), [3], [3'] (number of partitions of 3 into an odd number of distinct parts with 2 types of each part) and 4 - 2 = 2. Square array begins: 1, 1, 1, 1, 1, 1, ... 0, -1, -2, -3, -4, -5, ... 0, -1, -1, 0, 2, 5, ... 0, 0, 2, 5, 8, 10, ... 0, 0, 1, 0, -5, -15, ... 0, 1, 2, 0, -4, -6, ... MAPLE A:= proc(n, k) option remember; `if`(n=0, 1, -k* add(numtheory[sigma](j)*A(n-j, k), j=1..n)/n) end: seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Jun 21 2018 MATHEMATICA Table[Function[k, SeriesCoefficient[Product[(1 - x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[QPochhammer[x, x, Infinity]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[Sum[(-1)^i*x^(i*(3*i + 1)/2), {i, -Infinity, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten CROSSREFS Columns k=0-20 give: A000007, A010815, A002107, A010816, A000727, A000728, A000729, A000730, A000731, A010817, A010818, A010819, A000735, A010820, A010821, A010822, A000739, A010823, A010824, A010825, A010826. Main diagonal gives A008705. Antidiagonal sums give A299105. Sequence in context: A267181 A131185 A307819 * A296067 A306713 A303810 Adjacent sequences: A286351 A286352 A286353 * A286355 A286356 A286357 KEYWORD sign,tabl AUTHOR Ilya Gutkovskiy, May 08 2017 STATUS approved

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Last modified June 8 18:04 EDT 2023. Contains 363165 sequences. (Running on oeis4.)