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 A367468 #12 Nov 22 2023 22:29:51
%S A367468 0,1,0,2,4,0,3,17,9,0,4,52,68,16,0,5,139,345,190,25,0,6,346,1474,1440,
%T A367468 430,36,0,7,825,5733,8904,4550,847,49,0,8,1912,21048,49056,38304,
%U A367468 11928,1512,64,0,9,4343,74385,251250,282135,130998,27342,2508,81,0
%N A367468 Triangle read by rows: T(n,k) is the total number of movable letters in all members of the k-partitions of [n], with 1 <= k <= n.
%H A367468 Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.1080/10236198.2020.1739275">Counting set partitions by the number of movable letters</a>, Journal of Difference Equations and Applications, 26:3, 384-403, (2020). <a href="https://www.researchgate.net/publication/374910923_Counting_set_partitions_by_the_number_of_movable_letters">On ResearchGate</a>. See Theorem 8.
%F A367468 T(n,1) = n-1 and T(n,k) = (2*n - 1)*S2(n,k)/2 - S2(n+1,k)/2 + S2(n-1,k-2)/2 for k > 1.
%F A367468 T(n,2) = A208658(n-1).
%F A367468 T(n,n-1) = A000290(n-1) for n > 1.
%e A367468 Triangle begins:
%e A367468 0;
%e A367468 1, 0;
%e A367468 2, 4, 0;
%e A367468 3, 17, 9, 0;
%e A367468 4, 52, 68, 16, 0;
%e A367468 5, 139, 345, 190, 25, 0;
%e A367468 ...
%t A367468 T[n_,k_]:=If[k==1,n-1,(2n-1)StirlingS2[n,k]/2-StirlingS2[n+1,k]/2+StirlingS2[n-1,k-2]/2]; Table[T[n,k],{n,10},{k,n}]//Flatten
%Y A367468 Cf. A000290, A008277, A208658, A367469 (row sums).
%K A367468 nonn,tabl
%O A367468 1,4
%A A367468 _Stefano Spezia_, Nov 19 2023