A306376
Sum of the 2 X 2 minors in the n X n Pascal matrix.
Original entry on oeis.org
0, 0, 1, 7, 34, 144, 574, 2226, 8533, 32587, 124453, 476145, 1826175, 7022379, 27072487, 104614863, 405122290, 1571859864, 6109296442, 23781666198, 92704406320, 361832294964, 1413879679672, 5530590849168, 21654384302110, 84859670743770, 332818903663390
Offset: 0
-
a:= proc(n) option remember; `if`(n<3, (n-1)*n/2,
((7*n^2-16*n+6)*a(n-1)-2*(7*n^2-17*n+9)*a(n-2)+
4*(n-1)*(2*n-3)*a(n-3))/(n*(n-2)))
end:
seq(a(n), n=0..30);
-
a[n_] := a[n] = If[n < 3, (n-1)n/2,
((7n^2 - 16n + 6) a[n-1] - 2(7n^2 - 17n + 9) a[n-2] +
4(n-1)(2n-3) a[n-3])/(n(n-2))];
a /@ Range[0, 30] (* Jean-François Alcover, May 03 2021, after Alois P. Heinz *)
A369559
T(n,k) is the sum of the permanents of all k X k submatrices in the n X n Pascal matrix; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 7, 9, 1, 1, 15, 50, 35, 1, 1, 31, 234, 482, 185, 1, 1, 63, 1016, 5011, 6894, 1267, 1, 1, 127, 4256, 46252, 162724, 150624, 10633, 1, 1, 255, 17509, 403316, 3231672, 8369812, 4900141, 105219, 1, 1, 511, 71349, 3415771, 59157822, 362855438, 696003275, 223813933, 1196889, 1
Offset: 0
T(3,2) = 9:
The 3 X 3 Pascal matrix
[1 0 0]
[1 1 0]
[1 2 1]
has nine 2 X 2 submatrices
[1 0] [1 0] [0 0] [1 0] [1 0] [0 0] [1 1] [1 0] [1 0]
[1 1] [1 0] [1 0] [1 2] [1 1] [2 1] [1 2] [1 1] [2 1].
Sum of their permanents is 1 + 0 + 0 + 2 + 1 + 0 + 3 + 1 + 1 = 9.
Triangle T(n,k) begins:
1;
1, 1;
1, 3, 1;
1, 7, 9, 1;
1, 15, 50, 35, 1;
1, 31, 234, 482, 185, 1;
1, 63, 1016, 5011, 6894, 1267, 1;
1, 127, 4256, 46252, 162724, 150624, 10633, 1;
1, 255, 17509, 403316, 3231672, 8369812, 4900141, 105219, 1;
...
-
with(combinat): with(LinearAlgebra):
T:= proc(n, k) option remember; `if`(k=0 or k=n, 1, (l-> add(add(
Permanent(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..9);
-
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[Permanent /@ Flatten[Table[M[[i, j]], {i, l}, {j, l}], 1]]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 29 2024 *)
Showing 1-2 of 2 results.