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

%I A382225 #37 Mar 22 2025 17:39:39
%S A382225 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,
%T A382225 1,1,28,266,798,871,346,43,1,1,36,462,1974,3321,2306,631,57,1,1,45,
%U A382225 750,4326,10377,11126,5335,1065,73,1,1,55,1155,8646,28017,42878,31795,11145,1693,91,1
%N A382225 Triangle read by rows: T(n,k) = Sum_{i=k..n} C(i-1,i-k)*C(i,k).
%C A382225 Triangle T(n,k) of minors of the main diagonal of Pascal's matrix, n -size matrix, k - number of element of diagonal.
%H A382225 Paolo Xausa, <a href="/A382225/b382225.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%H A382225 Wikipedia, <a href="https://en.wikipedia.org/wiki/Minor_(linear_algebra)">Minor (linear algebra)</a>
%F A382225 G.f.: 1/(1-x) * ((1-x*(1-y))/(2*(sqrt((1-x*(1+y))^2-4*x^2*y)))+1/2).
%F A382225 T(n,k) = T(n-1,k)+C(n-1,k-1)*C(n,k).
%e A382225 Triangle starts:
%e A382225   1;
%e A382225   1,  1;
%e A382225   1,  3,   1;
%e A382225   1,  6,   7,   1;
%e A382225   1, 10,  25,  13,   1;
%e A382225   1, 15,  65,  73,  21,  1;
%e A382225   1, 21, 140, 273, 171, 31, 1;
%e A382225   ...
%p A382225 T:= proc(n, k) option remember; `if`(n<0, 0,
%p A382225       T(n-1, k)+binomial(n-1, k-1)*binomial(n, k))
%p A382225     end:
%p A382225 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Mar 20 2025
%t A382225 A382225[n_, k_] := A382225[n, k] = If[k == n, 1, A382225[n-1, k] + Binomial[n-1, k-1]*Binomial[n, k]];
%t A382225 Table[A382225[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Mar 22 2025 *)
%o A382225 (Maxima)
%o A382225 h[i,j]:=binomial(i+j-3,i-1);
%o A382225 for n:1 thru 7 do
%o A382225     if n=1 then print([1])
%o A382225     else (M:genmatrix(h,n,n),
%o A382225           print(makelist(determinant(minor(M,k,k)),k,1,n))
%o A382225          );
%Y A382225 Columns k=0-3 give: A000012, A000217, A001296(n-1) for n>=1, A107963(n-3) for n>=3.
%Y A382225 Row sums give A024718.
%Y A382225 T(n+1,n) gives A002061(n+1).
%Y A382225 Cf. A007318, A184173.
%K A382225 nonn,tabl
%O A382225 0,5
%A A382225 _Vladimir Kruchinin_, Mar 19 2025