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A242882 Number of compositions of n into parts with distinct multiplicities.

%I A242882 #24 Aug 31 2019 22:25:58
%S A242882 1,1,2,2,6,12,16,40,60,82,216,538,788,2034,3740,6320,13336,27498,
%T A242882 42936,93534,173520,351374,734650,1592952,3033194,6310640,12506972,
%U A242882 25296110,49709476,101546612,195037028,391548336,764947954,1527004522,2953533640,5946359758
%N A242882 Number of compositions of n into parts with distinct multiplicities.
%H A242882 Vaclav Kotesovec, <a href="/A242882/b242882.txt">Table of n, a(n) for n = 0..264</a> (terms 0..200 from Alois P. Heinz)
%H A242882 Vaclav Kotesovec, <a href="/A242882/a242882.jpg">What is the limit a(n)/2^n ?</a>
%e A242882 a(0) = 1: the empty composition.
%e A242882 a(1) = 1: [1].
%e A242882 a(2) = 2: [1,1], [2].
%e A242882 a(3) = 2: [1,1,1], [3].
%e A242882 a(4) = 6: [1,1,1,1], [1,1,2], [1,2,1], [2,1,1], [2,2], [4].
%e A242882 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].
%p A242882 b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
%p A242882       `if`(i<1, 0, add(`if`(j>0 and j in s, 0,
%p A242882       b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i)))
%p A242882     end:
%p A242882 a:= n-> b(n$2, {}):
%p A242882 seq(a(n), n=0..45);
%t A242882 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}]]];
%t A242882 a[n_] := b[n, n, {}];
%t A242882 Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%o A242882 (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
%Y A242882 Row sums of A242887 and of A242896.
%Y A242882 Cf. A098859 (the same for partitions).
%K A242882 nonn
%O A242882 0,3
%A A242882 _Alois P. Heinz_, May 25 2014