0,9

Alois P. Heinz, Rows n = 0..20, flattened

T(n,k) = A276719(n,k) - A276719(n,k-1) for k>0, T(n,0) = A000007(n).

T(4,1) = 1: 1|2|3|4.

T(4,2) = 4: 12|34, 12|3|4, 1|23|4, 1|2|34.

T(4,3) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3.

T(4,4) = 5: 1234, 124|3, 134|2, 14|23, 14|2|3.

T(5,4) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.

Triangle T(n,k) begins:

1;

0, 1;

0, 1, 1;

0, 1, 2, 2;

0, 1, 4, 5, 5;

0, 1, 7, 12, 17, 15;

0, 1, 12, 29, 45, 64, 52;

0, 1, 20, 66, 121, 201, 265, 203;

0, 1, 33, 145, 336, 585, 966, 1197, 877;

...

b:= proc(n, m, l) option remember; `if`(n=0, 1,

add(b(n-1, max(m, j), [subsop(1=NULL, l)[],

`if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))

end:

A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):

T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):

seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)

Columns k=0-10 give: A000007, A057427, A000071(n+1), A320553, A320554, A320555, A320556, A320557, A320558, A320559, A320560.

Row sums give A000110.

Main diagonal gives A000110(n-1) for n>0.

T(2n,n) gives A276728.

Cf. A263757, A276719, A276891.

Sequence in context: A317575 A295653 A146326 * A267617 A158852 A188285

Adjacent sequences: A276724 A276725 A276726 * A276728 A276729 A276730

nonn,tabl

Alois P. Heinz, Sep 16 2016

approved