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A282988 Triangle of partitions of an n-set into boxes of size >= m.

%I A282988 #22 Sep 22 2023 12:26:51
%S A282988 1,2,1,5,1,1,15,4,1,1,52,11,1,1,1,203,41,11,1,1,1,877,162,36,1,1,1,1,
%T A282988 4140,715,92,36,1,1,1,1,21147,3425,491,127,1,1,1,1,1,115975,17722,
%U A282988 2557,337,127,1,1,1,1,1,678570,98253,11353,793,463,1,1,1,1,1,1
%N A282988 Triangle of partitions of an n-set into boxes of size >= m.
%H A282988 Alois P. Heinz, <a href="/A282988/b282988.txt">Rows n = 1..141, flattened</a>
%F A282988 T(n,m) = Sum_{i=0..n-m} C(n-1, i+m-1)*T(n-i-m, m).
%F A282988 E.g.f. m column of T(n,m) is exp(exp(x)-Sum_{k=0..m} 1/k!x^k).
%e A282988 Triangle T(n,m) begins:
%e A282988     1;
%e A282988     2,   1;
%e A282988     5,   1,   1;
%e A282988    15,   4,   1,   1;
%e A282988    52,  11,   1,   1,   1;
%e A282988   203,  41,  11,   1,   1,   1;
%e A282988   877, 162,  36,   1,   1,   1,   1;
%e A282988   ...
%p A282988 T:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A282988       T(n-j, k)*binomial(n-1, j-1), j=k..n))
%p A282988     end:
%p A282988 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Sep 28 2017
%t A282988 T[n_, m_] := T[n, m] = Which[Or[n == m, n == 0], 1, m == 0, 0, True, Sum[Binomial[n - 1, i + m - 1] T[n - i - m, m], {i, 0, n - m}]]; Table[T[n, m], {n, 11}, {m, n}] // Flatten (* _Michael De Vlieger_, Feb 26 2017 *)
%o A282988 (Maxima)
%o A282988 T(n,m):=if n=m or n=0 then 1 else if m=0 then 0 else sum(binomial(n-1, i+m-1)*T(n-i-m,m), i, 0, n-m);
%Y A282988 Cf. A000110, A000296, A006505, A057837, A057814, A182931, A260878.
%K A282988 nonn,tabl
%O A282988 1,2
%A A282988 _Vladimir Kruchinin_, Feb 26 2017