A242887 - OEIS

%I #23 Aug 31 2019 22:25:48

%S 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 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 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 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 Alois P. Heinz, <a href="/A242887/b242887.txt">Rows n = 0..140, flattened</a>

%e T(5,1) = 1: [5].

%e T(5,3) = 6: [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].

%e T(5,4) = 4: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1].

%e T(5,5) = 1: [1,1,1,1,1].

%e Triangle T(n,k) begins:

%e   1;

%e   0, 1;

%e   0, 1, 1;

%e   0, 1, 0,  1;

%e   0, 1, 1,  3,  1;

%e   0, 1, 0,  6,  4,  1;

%e   0, 1, 1,  4,  4,  5,  1;

%e   0, 1, 0,  9,  8, 15,  6,  1;

%e   0, 1, 1,  9,  5, 15, 21,  7,  1;

%e   0, 1, 0, 10,  8, 20,  6, 28,  8, 1;

%e   0, 1, 1, 12, 12,  6, 96, 42, 36, 9, 1;

%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, x^j*

%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_] := 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 (PARI)

%o 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 { for(n=0, 10, print(T(n))) } \\ _Andrew Howroyd_, Aug 31 2019

%Y Columns k=0-10 give: A000007, A057427, A059841 (for n>1), A321773, A321774, A321775, A321776, A321777, A321778, A321779, A321780.

%Y Row sums give A242882.

%Y T(2n,n) gives A321772.

%Y Cf. A242896.

%K nonn,tabl

%O 0,14

%A _Alois P. Heinz_, May 25 2014