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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).

%I A184173 #36 Feb 29 2024 06:56:50
%S A184173 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,
%T A184173 63,1,1,127,2226,8526,8526,2226,127,1,1,255,8533,60588,113832,60588,
%U A184173 8533,255,1,1,511,32587,424117,1444608,1444608,424117,32587,511,1
%N 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).
%C A184173 Apparently, the sum of the entries in row n is A005157(n).
%H A184173 Alois P. Heinz, <a href="/A184173/b184173.txt">Rows n = 0..12, flattened</a>
%H A184173 Wikipedia, <a href="https://en.wikipedia.org/wiki/Minor_(linear_algebra)">Minor (linear algebra)</a>
%F A184173 The triangle is symmetric: T(n,k) = T(n,n-k).
%e A184173 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.
%e A184173 Triangle starts:
%e A184173   1;
%e A184173   1,   1;
%e A184173   1,   3,    1;
%e A184173   1,   7,    7,    1;
%e A184173   1,  15,   34,   15,    1;
%e A184173   1,  31,  144,  144,   31,    1;
%e A184173   1,  63,  574, 1155,  574,   63,   1;
%e A184173   1, 127, 2226, 8526, 8526, 2226, 127, 1;
%e A184173   ...
%p A184173 with(combinat): with(LinearAlgebra):
%p A184173 T:= proc(n, k) option remember; `if`(n-k<k, T(n, n-k), (l-> add(add(
%p A184173       Determinant(SubMatrix(Matrix(n, (i, j)-> binomial(i-1, j-1)),
%p A184173        i, j)), j in l), i in l))(choose([$1..n], k)))
%p A184173     end:
%p A184173 seq(seq(T(n, k), k=0..n), n=0..7);  # _Alois P. Heinz_, Feb 11 2019
%t A184173 T[n_, k_] := T[n, k] = If[k == 0 || k == n, 1, Module[{l, M},
%t A184173     l = Subsets[Range[n], {k}];
%t A184173     M = Table[Binomial[i-1, j-1], {i, n}, {j, n}];
%t A184173     Total[Det /@ Flatten[Table[M[[i, j]], {i, l}, {j, l}], 1]]]];
%t A184173 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 09 2019 updated Feb 29 2024 *)
%Y A184173 Columns k=0-2 give: A000012, A000225, A306376.
%Y A184173 Cf. A005157, A007318.
%K A184173 nonn,tabl
%O A184173 0,5
%A A184173 _Emeric Deutsch_, Jan 12 2011
%E A184173 Typo corrected by _Alois P. Heinz_, Feb 11 2019