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A286897 Sum T(n,k) of the k-th last entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A286897 #16 Feb 02 2025 11:08:19
%S A286897 1,5,1,23,6,1,109,33,7,1,544,182,45,8,1,2876,1034,284,59,9,1,16113,
%T A286897 6122,1815,420,75,10,1,95495,37927,11931,2987,595,93,11,1,597155,
%U A286897 246030,81205,21620,4665,814,113,12,1,3929243,1669941,573724,160607,36900,6979,1082,135,13,1
%N A286897 Sum T(n,k) of the k-th last entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A286897 Alois P. Heinz, <a href="/A286897/b286897.txt">Row n = 1..50, flattened</a>
%H A286897 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A286897 T(3,2) = 6 because the sum of the second last entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 2+1+1+2 = 6.
%e A286897 Triangle T(n,k) begins:
%e A286897       1;
%e A286897       5,     1;
%e A286897      23,     6,     1;
%e A286897     109,    33,     7,    1;
%e A286897     544,   182,    45,    8,   1;
%e A286897    2876,  1034,   284,   59,   9,  1;
%e A286897   16113,  6122,  1815,  420,  75, 10,  1;
%e A286897   95495, 37927, 11931, 2987, 595, 93, 11, 1;
%e A286897   ...
%p A286897 b:= proc(n, l) option remember; `if`(n=0, [1, 0],
%p A286897       (p-> p+[0, n*p[1]*x^1])(b(n-1, [l[], 1]))+
%p A286897        add((p-> p+[0, n*p[1]*x^(l[j]+1)])(b(n-1,
%p A286897        sort(subsop(j=l[j]+1, l), `>`))), j=1..nops(l)))
%p A286897     end:
%p A286897 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, [])[2]):
%p A286897 seq(T(n), n=1..14);
%t A286897 b[0, _] = {1, 0}; b[n_, l_] := b[n, l] = Function[p, p + {0, n*p[[1]]*x^1} ][b[n - 1, Append[l, 1]]] + Sum[Function[p, p + {0, n*p[[1]]*x^(l[[j]] + 1)}][b[n - 1, Reverse @ Sort[ReplacePart[l, j -> l[[j]] + 1]]]], {j, 1, Length[l]}];
%t A286897 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, n}]][b[n, {}][[2]]];
%t A286897 Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, May 26 2018, from Maple *)
%Y A286897 Column k=1 gives A278677(n-1).
%Y A286897 Row sums give A000110(n) * A000217(n) = A105488(n+3).
%Y A286897 Cf. A285595.
%K A286897 nonn,tabl
%O A286897 1,2
%A A286897 _Alois P. Heinz_, May 15 2017