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A213177
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Number T(n,k) of parts in all partitions of n with largest multiplicity k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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14
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0, 0, 1, 0, 1, 2, 0, 3, 0, 3, 0, 3, 5, 0, 4, 0, 5, 6, 4, 0, 5, 0, 8, 9, 7, 5, 0, 6, 0, 10, 13, 13, 5, 6, 0, 7, 0, 13, 23, 14, 15, 6, 7, 0, 8, 0, 18, 30, 27, 16, 13, 7, 8, 0, 9, 0, 25, 44, 33, 30, 18, 15, 8, 9, 0, 10, 0, 30, 58, 55, 36, 34, 15, 17, 9, 10, 0, 11
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OFFSET
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0,6
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LINKS
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FORMULA
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EXAMPLE
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T(6,1) = 8: partitions of 6 with largest multiplicity 1 are [3,2,1], [4,2], [5,1], [6], with 3+2+2+1 = 8 parts.
T(6,2) = 9: [2,2,1,1], [3,3], [4,1,1].
T(6,3) = 7: [2,2,2], [3,1,1,1].
T(6,4) = 5: [2,1,1,1,1].
T(6,5) = 0.
T(6,6) = 6: [1,1,1,1,1,1].
Triangle begins:
0;
0, 1;
0, 1, 2;
0, 3, 0, 3;
0, 3, 5, 0, 4;
0, 5, 6, 4, 0, 5;
0, 8, 9, 7, 5, 0, 6;
0, 10, 13, 13, 5, 6, 0, 7;
0, 13, 23, 14, 15, 6, 7, 0, 8;
...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
end:
T:= (n, k)-> b(n, n, k)[2] -b(n, n, k-1)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[_, 0] = 0; T[n_, k_] := b[n, n, k][[2]] - b[n, n, k-1][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000004, A015723, A320372, A320373, A320374, A320375, A320376, A320377, A320378, A320379, A320380.
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KEYWORD
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AUTHOR
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STATUS
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approved
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