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A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A242887 #23 Aug 31 2019 22:25:48
%S A242887 1,0,1,0,1,1,0,1,0,1,0,1,1,3,1,0,1,0,6,4,1,0,1,1,4,4,5,1,0,1,0,9,8,15,
%T A242887 6,1,0,1,1,9,5,15,21,7,1,0,1,0,10,8,20,6,28,8,1,0,1,1,12,12,6,96,42,
%U A242887 36,9,1,0,1,0,15,12,30,192,168,64,45,10,1,0,1,1,13,9,20,142,238,204,93,55,11,1
%N A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A242887 Alois P. Heinz, <a href="/A242887/b242887.txt">Rows n = 0..140, flattened</a>
%e A242887 T(5,1) = 1: [5].
%e A242887 T(5,3) = 6: [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].
%e A242887 T(5,4) = 4: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1].
%e A242887 T(5,5) = 1: [1,1,1,1,1].
%e A242887 Triangle T(n,k) begins:
%e A242887   1;
%e A242887   0, 1;
%e A242887   0, 1, 1;
%e A242887   0, 1, 0,  1;
%e A242887   0, 1, 1,  3,  1;
%e A242887   0, 1, 0,  6,  4,  1;
%e A242887   0, 1, 1,  4,  4,  5,  1;
%e A242887   0, 1, 0,  9,  8, 15,  6,  1;
%e A242887   0, 1, 1,  9,  5, 15, 21,  7,  1;
%e A242887   0, 1, 0, 10,  8, 20,  6, 28,  8, 1;
%e A242887   0, 1, 1, 12, 12,  6, 96, 42, 36, 9, 1;
%p A242887 b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
%p A242887       `if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, x^j*
%p A242887        b(n-i*j, i-1,`if`(j=0, s, s union {j}))/j!), j=0..n/i))))
%p A242887     end:
%p A242887 T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2, {})):
%p A242887 seq(T(n), n=0..16);
%t A242887 b[n_, i_, s_] := b[n, i, s] = If[n==0, Total[s]!, If[i<1, 0, Expand[Sum[ If[j>0 && MemberQ[s, j], 0, x^j*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 08 2017, translated from Maple *)
%o A242887 (PARI)
%o A242887 T(n)={Vecrev(((r,k,b,w)->if(!k||!r, if(r,0,w!*x^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))}
%o A242887 { for(n=0, 10, print(T(n))) } \\ _Andrew Howroyd_, Aug 31 2019
%Y A242887 Columns k=0-10 give: A000007, A057427, A059841 (for n>1), A321773, A321774, A321775, A321776, A321777, A321778, A321779, A321780.
%Y A242887 Row sums give A242882.
%Y A242887 T(2n,n) gives A321772.
%Y A242887 Cf. A242896.
%K A242887 nonn,tabl
%O A242887 0,14
%A A242887 _Alois P. Heinz_, May 25 2014