%I #102 Apr 21 2026 16:14:21

%S 1,0,1,0,1,0,1,3,0,1,6,0,1,25,0,1,60,60,0,1,231,210,0,1,658,1400,0,1,

%T 2619,7560,0,1,8550,40320,12600,0,1,35695,224070,69300,0,1,129756,

%U 1315380,693000,0,1,568503,7610460,5405400,0,1,2255344,46766720,46666620

%N Triangle T(n,k) read by rows: T(n,k) is the number of partitions of [n] whose set of block sizes equals [k]; n >= 0, 0 <= k <= A003056(n).

%H Alois P. Heinz, <a href="/A387082/b387082.txt">Rows n = 0..560, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F T(A000217(n),n) = A022915(n).

%e T(5,2) = 25: 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 15|23|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1;

%e 0, 1, 3;

%e 0, 1, 6;

%e 0, 1, 25;

%e 0, 1, 60, 60;

%e 0, 1, 231, 210;

%e 0, 1, 658, 1400;

%e 0, 1, 2619, 7560;

%e 0, 1, 8550, 40320, 12600;

%e 0, 1, 35695, 224070, 69300;

%e 0, 1, 129756, 1315380, 693000;

%e ...

%p f:= proc(n) option remember; floor((sqrt(1+8*n)-1)/2) end:

%p T:= proc(n, i) option remember; `if`(k=0, 1-signum(n), `if`(n=0,

%p `if`(i=0, 1, 0), add(T(n-i*j, i-1)*combinat[multinomial]

%p (n, n-i*j, i$j)/j!, j=`if`(i=1, n, 1..n/i))))

%p end:

%p seq(seq(T(n,k), k=0..f(n)), n=0..14);

%Y Row sums give A393098.

%Y Last terms of rows give A394281.

%Y Cf. A000217, A003056, A022915.

%K nonn,tabf

%O 0,8

%A _Alois P. Heinz_, Apr 17 2026