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A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A372762 #18 May 12 2024 14:54:59
%S A372762 1,0,1,0,1,1,0,4,0,1,0,5,9,0,1,0,31,10,10,0,1,0,82,70,35,15,0,1,0,344,
%T A372762 336,140,35,21,0,1,0,1661,1393,616,385,56,28,0,1,0,7942,6210,4984,
%U A372762 1386,504,84,36,0,1,0,38721,41331,22590,8610,3717,840,120,45,0,1
%N A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A372762 Alois P. Heinz, <a href="/A372762/b372762.txt">Rows n = 0..140, flattened</a>
%H A372762 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A372762 Sum_{k=0..n} k * T(n,k) = A372650(n).
%e A372762 T(5,1) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
%e A372762 T(5,2) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
%e A372762 T(5,3) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
%e A372762 T(5,4) = 0.
%e A372762 T(5,5) = 1: 1|2|3|4|5.
%e A372762 Triangle T(n,k) begins:
%e A372762   1;
%e A372762   0,     1;
%e A372762   0,     1,     1;
%e A372762   0,     4,     0,     1;
%e A372762   0,     5,     9,     0,    1;
%e A372762   0,    31,    10,    10,    0,    1;
%e A372762   0,    82,    70,    35,   15,    0,   1;
%e A372762   0,   344,   336,   140,   35,   21,   0,   1;
%e A372762   0,  1661,  1393,   616,  385,   56,  28,   0,  1;
%e A372762   0,  7942,  6210,  4984, 1386,  504,  84,  36,  0, 1;
%e A372762   0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1;
%e A372762   ...
%p A372762 b:= proc(n, m, t) option remember; `if`(n=0, x^t,
%p A372762       add(binomial(n-1, j-1)*b(n-j, min(j, m),
%p A372762      `if`(j<m, 1, `if`(j=m, t+1, t))), j=1..n))
%p A372762     end:
%p A372762 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
%p A372762 seq(T(n), n=0..12);
%Y A372762 Columns k=0-2 give: A000007, A224219, A372764.
%Y A372762 Row sums give A000110.
%Y A372762 T(2n,n) gives A271425.
%Y A372762 Cf. A372650, A372722.
%K A372762 nonn,tabl
%O A372762 0,8
%A A372762 _Alois P. Heinz_, May 12 2024