This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A283424 #37 May 17 2023 15:14:14
%S A283424 1,2,1,5,3,1,15,10,4,1,52,37,17,5,1,203,151,76,26,6,1,877,674,362,137,
%T A283424 37,7,1,4140,3263,1842,750,225,50,8,1,21147,17007,9991,4307,1395,345,
%U A283424 65,9,1,115975,94828,57568,25996,8944,2392,502,82,10,1
%N A283424 Number T(n,k) of blocks of size >= k in all set partitions of [n], assuming that every set partition contains one block of size zero; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A283424 T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.
%H A283424 Alois P. Heinz, <a href="/A283424/b283424.txt">Rows n = 0..140, flattened</a>
%H A283424 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A283424 T(n,k) = Sum_{j=0..n-k} binomial(n,j) * Bell(j).
%F A283424 T(n,k) = Bell(n+1) - Sum_{j=0..k-1} binomial(n,j) * Bell(n-j).
%F A283424 T(n,k) = Sum_{j=k..n} A056857(n+1,j) = Sum_{j=k..n} A056860(n+1,n+1-j).
%F A283424 Sum_{k=0..n} T(n,k) = n*Bell(n)+Bell(n+1) = A124427(n+1).
%F A283424 Sum_{k=1..n} T(n,k) = n*Bell(n) = A070071(n).
%F A283424 T(n,0)-T(n,1) = Bell(n).
%F A283424 Sum_{k=0..n} (-1)^k * T(n,k) = A224271(n+1). - _Alois P. Heinz_, May 17 2023
%e A283424 T(3,2) = 4 because the number of blocks of size >= 2 in all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+1+1+1+0 = 4.
%e A283424 Triangle T(n,k) begins:
%e A283424 1;
%e A283424 2, 1;
%e A283424 5, 3, 1;
%e A283424 15, 10, 4, 1;
%e A283424 52, 37, 17, 5, 1;
%e A283424 203, 151, 76, 26, 6, 1;
%e A283424 877, 674, 362, 137, 37, 7, 1;
%e A283424 4140, 3263, 1842, 750, 225, 50, 8, 1;
%e A283424 21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1;
%e A283424 ...
%p A283424 T:= proc(n, k) option remember; `if`(k>n, 0,
%p A283424 binomial(n, k)*combinat[bell](n-k)+T(n, k+1))
%p A283424 end:
%p A283424 seq(seq(T(n, k), k=0..n), n=0..14);
%t A283424 T[n_, k_] := Sum[Binomial[n, j]*BellB[j], {j, 0, n - k}];
%t A283424 Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2018 *)
%Y A283424 Columns k=0-10 give: A000110(n+1), A138378 or A005493(n-1), A124325, A288785, A288786, A288787, A288788, A288789, A288790, A288791, A288792.
%Y A283424 Row sums give A124427(n+1).
%Y A283424 T(2n,n) gives A286896.
%Y A283424 Cf. A005493, A056857, A056860, A070071, A224271, A285595.
%K A283424 nonn,tabl
%O A283424 0,2
%A A283424 _Alois P. Heinz_, May 14 2017