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 A134979 Triangle read by rows: T(n,k) = number of partitions of n where the maximum number of objects in partitions of any given size is k. 2
 1, 0, 2, 0, 1, 2, 0, 1, 1, 3, 0, 0, 3, 2, 2, 0, 0, 2, 4, 1, 4, 0, 0, 1, 6, 3, 3, 2, 0, 0, 1, 6, 4, 6, 1, 4, 0, 0, 0, 6, 7, 8, 3, 3, 3, 0, 0, 0, 5, 7, 14, 4, 6, 2, 4, 0, 0, 0, 5, 7, 18, 7, 9, 5, 3, 2, 0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6, 0, 0, 0, 2, 9, 26, 15, 19, 11, 9, 3, 5, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Every column is eventually 0; the last row with a nonzero value in column k is A024916(k). T(A024916(k)-i, k) <= P(i), where P is the partition function (A000041); equality holds for 0 <= i <= k. The partition represented by the last number in column k is row k of A010766. LINKS Alois P. Heinz, Rows n = 1..200, flattened EXAMPLE For the partition [3,2^2], there are 3 objects in the part of size 3 and 4 objects in the parts of size 2, so this partition is counted towards T(7,4). Triangle T(n,k) begins: 1; 0, 2; 0, 1, 2; 0, 1, 1, 3; 0, 0, 3, 2, 2; 0, 0, 2, 4, 1, 4; 0, 0, 1, 6, 3, 3, 2; 0, 0, 1, 6, 4, 6, 1, 4; 0, 0, 0, 6, 7, 8, 3, 3, 3; 0, 0, 0, 5, 7, 14, 4, 6, 2, 4; 0, 0, 0, 5, 7, 18, 7, 9, 5, 3, 2; 0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6; ... MAPLE b:= proc(n, i, m) option remember; `if`(n=0 or i=1, x^ max(m, n), add(b(n-i*j, i-1, max(m, i*j)), j=0..n/i)) end: T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n\$2, 0)): seq(T(n), n=1..20); # Alois P. Heinz, Feb 07 2020 MATHEMATICA b[n_, i_, m_] := b[n, i, m] = If[n == 0 || i == 1, x^Max[m, n], Sum[b[n - i j, i - 1, Max[m, i j]], {j, 0, n/i}]]; T[n_] := Table[Coefficient[b[n, n, 0], x, i], {i, 1, n}]; Array[T, 20] // Flatten (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *) CROSSREFS Cf. A008284, A091602, A000041 (row sums), A000005 (main diagonal), A032741 (2nd diagonal), A010766. Column sums give A332233. Sequence in context: A227698 A331466 A166124 * A112248 A244860 A308009 Adjacent sequences: A134976 A134977 A134978 * A134980 A134981 A134982 KEYWORD nonn,look,tabl AUTHOR Franklin T. Adams-Watters, Feb 04 2008 STATUS approved

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Last modified May 30 15:53 EDT 2024. Contains 372968 sequences. (Running on oeis4.)