%I #26 Apr 30 2020 11:33:10
%S 1,1,1,2,1,1,5,4,3,1,15,11,9,4,1,52,41,35,20,5,1,203,162,150,90,30,6,
%T 1,877,715,672,455,175,42,7,1,4140,3425,3269,2352,1015,280,56,8,1,
%U 21147,17722,17271,13132,6237,1890,420,72,9,1,115975,98253,97155,76540,39480,12978,3150,600,90,10,1
%N Number T(n,k) of set partitions of [n] such that at least one of the block sizes is k or k=0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A327884/b327884.txt">Rows n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F E.g.f. of column k: exp(exp(x)-1) - [k>0] * exp(exp(x)-1-x^k/k!).
%F T(n,0) - T(n,1) = A000296(n).
%e T(4,1) = 11: 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
%e T(4,2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
%e T(4,3) = 4: 123|4, 124|3, 134|2, 1|234.
%e T(4,4) = 1: 1234.
%e T(5,1) = 41: 1234|5, 1235|4, 123|4|5, 1245|3, 124|3|5, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 12|3|4|5, 1345|2, 134|2|5, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
%e Triangle T(n,k) begins:
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 5, 4, 3, 1;
%e 15, 11, 9, 4, 1;
%e 52, 41, 35, 20, 5, 1;
%e 203, 162, 150, 90, 30, 6, 1;
%e 877, 715, 672, 455, 175, 42, 7, 1;
%e 4140, 3425, 3269, 2352, 1015, 280, 56, 8, 1;
%e 21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1;
%e ...
%p b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
%p end:
%p T:= (n, k)-> b(n, 0)-`if`(k=0, 0, b(n, k)):
%p seq(seq(T(n, k), k=0..n), n=0..11);
%t b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[If[j == k, 0, b[n - j,k] Binomial[ n - 1, j - 1]], {j, 1, n}]];
%t T[n_, k_] := b[n, 0] - If[k == 0, 0, b[n, k]];
%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-3 give: A000110, A000296(n+1), A327885, A328153.
%Y T(2n,n) gives A276961.
%Y Cf. A080510, A327869.
%K nonn,tabl
%O 0,4
%A _Alois P. Heinz_, Sep 28 2019