%I #31 Jun 06 2024 08:35:20

%S 1,0,1,0,1,0,1,2,0,1,3,0,1,7,0,1,11,6,0,1,20,12,0,1,32,32,0,1,54,72,0,

%T 1,87,152,24,0,1,143,311,60,0,1,231,625,180,0,1,376,1225,450,0,1,608,

%U 2378,1116,0,1,986,4566,2544,120,0,1,1595,8700,5752,360

%N Number T(n,k) of compositions of n such that the set of parts is [k]; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

%H Alois P. Heinz, <a href="/A373118/b373118.txt">Rows n = 0..750, flattened</a>

%F T(A000217(n),n) = n! = A000142(n).

%F T(A000124(n),n) = A001710(n+1) for n>=1.

%F T(A000290(n),n) = T(n^2,n) = A332721(n).

%F G.f. for column k: C({1..k},x) where C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/ (1 - Sum_{i in {s}} (x^i)) with C({},x) = 1. - _John Tyler Rascoe_, May 25 2024

%e T(6,2) = 11: 1122, 1212, 1221, 2112, 2121, 2211, 11112, 11121, 11211, 12111, 21111.

%e T(7,3) = 12: 1123, 1132, 1213, 1231, 1312, 1321, 2113, 2131, 2311, 3112, 3121, 3211.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1;

%e 0, 1, 2;

%e 0, 1, 3;

%e 0, 1, 7;

%e 0, 1, 11, 6;

%e 0, 1, 20, 12;

%e 0, 1, 32, 32;

%e 0, 1, 54, 72;

%e 0, 1, 87, 152, 24;

%e 0, 1, 143, 311, 60;

%e 0, 1, 231, 625, 180;

%e 0, 1, 376, 1225, 450;

%e 0, 1, 608, 2378, 1116;

%e 0, 1, 986, 4566, 2544, 120;

%e ...

%p b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0),

%p `if`(i<1 or n<i*(i+1)/2, 0, add(b(n-i*j, i-1, t+j)/j!, j=1..n/i)))

%p end:

%p T:= (n, k)-> b(n, k, 0):

%p seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..18);

%Y Columns k=0-3 give: A000007, A057427, A245738, A372702.

%Y Row sums give A107429.

%Y Cf. A000124, A000142, A000217, A000290, A001710, A003056, A008289, A332721, A371417, A373305, A373306.

%K nonn,look,tabf

%O 0,8

%A _Alois P. Heinz_, May 25 2024