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A327884 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.

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