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

0,5

Comments

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

Links

Alois P. Heinz, Rows n = 0..12, flattened

Wikipedia, Minor (linear algebra)

Formula

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

Example

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;
  ...

Maple

with(combinat): with(LinearAlgebra):
T:= proc(n, k) option remember; `if`(n-k<k, T(n, n-k), (l-> 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 *)

Crossrefs

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

Cf. A005157, A007318.

Sequence in context: A157152 A136126 A046802 * A359985 A022166 A141689

Adjacent sequences: A184170 A184171 A184172 * A184174 A184175 A184176

Keyword

nonn,tabl

Author

Emeric Deutsch, Jan 12 2011

Extensions

Typo corrected by Alois P. Heinz, Feb 11 2019

Status

approved