A382225 Triangle read by rows: T(n,k) = Sum_{i=k..n} C(i-1,i-k)*C(i,k).

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 25, 13, 1, 1, 15, 65, 73, 21, 1, 1, 21, 140, 273, 171, 31, 1, 1, 28, 266, 798, 871, 346, 43, 1, 1, 36, 462, 1974, 3321, 2306, 631, 57, 1, 1, 45, 750, 4326, 10377, 11126, 5335, 1065, 73, 1, 1, 55, 1155, 8646, 28017, 42878, 31795, 11145, 1693, 91, 1

Offset: 0

Author

Vladimir Kruchinin, Mar 19 2025

Keywords

• nonn
• tabl

Comments

Triangle T(n,k) of minors of the main diagonal of Pascal's matrix, n -size matrix, k - number of element of diagonal.

Examples

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,   1;
  1, 10,  25,  13,   1;
  1, 15,  65,  73,  21,  1;
  1, 21, 140, 273, 171, 31, 1;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
• Wikipedia, Minor (linear algebra)

Crossrefs

Columns k=0-3 give: A000012, A000217, A001296(n-1) for n>=1, A107963(n-3) for n>=3.

Row sums give A024718.

T(n+1,n) gives A002061(n+1).

Cf. A007318, A184173.

Programs

Maple

T:= proc(n, k) option remember; `if`(n<0, 0,
      T(n-1, k)+binomial(n-1, k-1)*binomial(n, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 20 2025

Mathematica

A382225[n_, k_] := A382225[n, k] = If[k == n, 1, A382225[n-1, k] + Binomial[n-1, k-1]*Binomial[n, k]];
Table[A382225[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 22 2025 *)

Maxima

h[i,j]:=binomial(i+j-3,i-1);
for n:1 thru 7 do
    if n=1 then print([1])
    else (M:genmatrix(h,n,n),
          print(makelist(determinant(minor(M,k,k)),k,1,n))
         );

Formula

G.f.: 1/(1-x) * ((1-x*(1-y))/(2*(sqrt((1-x*(1+y))^2-4*x^2*y)))+1/2).

T(n,k) = T(n-1,k)+C(n-1,k-1)*C(n,k).