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.

1, 5, 1, 23, 6, 1, 109, 33, 7, 1, 544, 182, 45, 8, 1, 2876, 1034, 284, 59, 9, 1, 16113, 6122, 1815, 420, 75, 10, 1, 95495, 37927, 11931, 2987, 595, 93, 11, 1, 597155, 246030, 81205, 21620, 4665, 814, 113, 12, 1, 3929243, 1669941, 573724, 160607, 36900, 6979, 1082, 135, 13, 1

Offset

1,2

Links

Alois P. Heinz, Row n = 1..50, flattened

Wikipedia, Partition of a set

Example

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.
Triangle T(n,k) begins:
:     1;
:     5,     1;
:    23,     6,     1;
:   109,    33,     7,    1;
:   544,   182,    45,    8,   1;
:  2876,  1034,   284,   59,   9,  1;
: 16113,  6122,  1815,  420,  75, 10,  1;
: 95495, 37927, 11931, 2987, 595, 93, 11, 1;

Maple

b:= proc(n, l) option remember; `if`(n=0, [1, 0],
      (p-> p+[0, n*p[1]*x^1])(b(n-1, [l[], 1]))+
       add((p-> p+[0, n*p[1]*x^(l[j]+1)])(b(n-1,
       sort(subsop(j=l[j]+1, l), `>`))), j=1..nops(l)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, [])[2]):
seq(T(n), n=1..14);

Mathematica

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[n_] := Function[p, Table[Coefficient[p, x, i], {i, n}]][b[n, {}][[2]]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 26 2018, from Maple *)

Crossrefs

Column k=1 gives A278677(n+1).

Row sums give A000110(n) * A000217(n) = A105488(n+3).

Cf. A285595.

Sequence in context: A347487 A213118 A259682 * A167572 A147476 A146675

Adjacent sequences: A286894 A286895 A286896 * A286898 A286899 A286900

Keyword

nonn,tabl

Author

Alois P. Heinz, May 15 2017

Status

approved