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%I #26 Jan 06 2019 18:40:02
%S 1,0,1,0,2,0,2,0,3,3,0,2,10,0,4,12,0,2,38,0,4,56,0,3,79,0,4,152,60,0,
%T 2,251,285,0,6,284,498,0,2,594,1438,0,4,920,2816,0,4,1108,5208,0,5,
%U 2136,11195,0,2,3402,24094,0,6,4407,38523,0,2,8350,85182
%N Number T(n,k) of compositions of n into k parts with distinct multiplicities, where parts are counted without multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.
%H Alois P. Heinz, <a href="/A242896/b242896.txt">Rows n = 0..200, flattened</a>
%e T(5,1) = 2: [1,1,1,1,1], [5].
%e T(5,2) = 10: [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].
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 2;
%e 0, 2;
%e 0, 3, 3;
%e 0, 2, 10;
%e 0, 4, 12;
%e 0, 2, 38;
%e 0, 4, 56;
%e 0, 3, 79;
%e 0, 4, 152, 60;
%p b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
%p `if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, `if`(j=0, 1, x)*
%p b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, {})):
%p seq(T(n), n=0..16);
%t b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n-i*j, i-1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 16}] // Flatten (* _Jean-François Alcover_, Feb 11 2015, after _Alois P. Heinz_ *)
%Y Columns k=0-7 give: A000007, A000005, A242900, A246230, A246231, A246232, A246233, A246234.
%Y Row sums give A242882.
%Y Cf. A182485 (the same for partitions), A242887.
%K nonn,tabf
%O 0,5
%A _Alois P. Heinz_, May 25 2014
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