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A242882
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Number of compositions of n into parts with distinct multiplicities.
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64
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1, 1, 2, 2, 6, 12, 16, 40, 60, 82, 216, 538, 788, 2034, 3740, 6320, 13336, 27498, 42936, 93534, 173520, 351374, 734650, 1592952, 3033194, 6310640, 12506972, 25296110, 49709476, 101546612, 195037028, 391548336, 764947954, 1527004522, 2953533640, 5946359758
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OFFSET
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0,3
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LINKS
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EXAMPLE
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a(0) = 1: the empty composition.
a(1) = 1: [1].
a(2) = 2: [1,1], [2].
a(3) = 2: [1,1,1], [3].
a(4) = 6: [1,1,1,1], [1,1,2], [1,2,1], [2,1,1], [2,2], [4].
a(5) = 12: [1,1,1,1,1], [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1], [5].
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MAPLE
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b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
`if`(i<1, 0, add(`if`(j>0 and j in s, 0,
b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i)))
end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..45);
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MATHEMATICA
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b[n_, i_, s_] := b[n, i, s] = If[n == 0, Sum[j, {j, s}]!, If[i < 1, 0, Sum[If[j > 0 && MemberQ[s, j], 0, b[n - i*j, i - 1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, {}];
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PROG
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(PARI) a(n)={((r, k, b, w)->if(!k||!r, if(r, 0, w!), sum(m=0, r\k, if(!m || !bittest(b, m), self()(r-k*m, k-1, bitor(b, 1<<m), w+m)/m!))))(n, n, 1, 0)} \\ Andrew Howroyd, Aug 31 2019
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CROSSREFS
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Cf. A098859 (the same for partitions).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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