A184173 Triangle read by rows: T(n,k) is the sum of the k X k minors in the n X n Pascal matrix (0<=k<=n; the empty 0 X 0 minor is defined to be 1).

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 34, 15, 1, 1, 31, 144, 144, 31, 1, 1, 63, 574, 1155, 574, 63, 1, 1, 127, 2226, 8526, 8526, 2226, 127, 1, 1, 255, 8533, 60588, 113832, 60588, 8533, 255, 1, 1, 511, 32587, 424117, 1444608, 1444608, 424117, 32587, 511, 1

Offset: 0

Author

Emeric Deutsch, Jan 12 2011

Keywords

• nonn
• tabl

Comments

Apparently, the sum of the entries in row n is A005157(n).

Examples

			T(3,1) = 7 because in the 3 X 3 Pascal matrix [1,0,0/1,1,0/1,2,1] the sum of the entries is 7.
Triangle starts:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,    1;
  1,  15,   34,   15,    1;
  1,  31,  144,  144,   31,    1;
  1,  63,  574, 1155,  574,   63,   1;
  1, 127, 2226, 8526, 8526, 2226, 127, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..12, flattened
• Wikipedia, Minor (linear algebra)

Crossrefs

Columns k=0-2 give: A000012, A000225, A306376.

Cf. A005157, A007318.

Programs

Maple

with(combinat): with(LinearAlgebra):
T:= proc(n, k) option remember; `if`(n-k add(add(
      Determinant(SubMatrix(Matrix(n, (i, j)-> binomial(i-1, j-1)),
       i, j)), j in l), i in l))(choose([$1..n], k)))
    end:
seq(seq(T(n, k), k=0..n), n=0..7);  # Alois P. Heinz, Feb 11 2019

Mathematica

T[n_, k_] := T[n, k] = If[k == 0 || k == n, 1, Module[{l, M},
    l = Subsets[Range[n], {k}];
    M = Table[Binomial[i-1, j-1], {i, n}, {j, n}];
    Total[Det /@ Flatten[Table[M[[i, j]], {i, l}, {j, l}], 1]]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2019 updated Feb 29 2024 *)

Formula

The triangle is symmetric: T(n,k) = T(n,n-k).

Extensions

Typo corrected by Alois P. Heinz, Feb 11 2019