A292865 - OEIS

%I #18 Oct 02 2017 01:01:57

%S 1,0,-1,8,-71,656,-4816,1920,168784,43920880,-3315147449,209095006856,

%T -19095123359744,1814464114046976,320005209305667584,

%U -253215321875947192320,-3298397219599339984896,24417272707694829159671808,265094852554176756050442657024,-931723550682987095264656018072440

%N Determinants of the symmetric matrices whose entries on and below the main diagonal correspond to those of Pascal's triangle.

%F a(n) = det(L_n+U_n-I_{n+1}), where L_n is the lower triangular Pascal matrix of order n, U_n is the transpose of L_n and I_n is the identity matrix of order n. Note that L_n, U_n and I_n all have determinant 1 for all n.

%e a(0) is the determinant of the 1 X 1 matrix whose sole entry is one.

%e a(1) is the determinant of the 2 X 2 matrix of all ones.

%e a(2) is the determinant of the 3 X 3 matrix

%e [1 1 1]

%e [1 1 2]

%e [1 2 1].

%e a(3) is the determinant of the 4 X 4 matrix

%e [1 1 1 1]

%e [1 1 2 3]

%e [1 2 1 3]

%e [1 3 3 1].

%t PascalMatrix = Function[n, Table[Table[Binomial[m, i], {i, 0, n}], {m, 0, n}]];

%t PascalDet = Function[n, Det[PascalMatrix[n] + Transpose[PascalMatrix[n]] - IdentityMatrix[n + 1]]];

%t Table[PascalDet[i], {i, 0, 19}]

%o (Python)

%o from sympy import *

%o def m(N):

%o     return Matrix([

%o         ([binomial(i, n) for n in range(i+1)] +[0] * (N-i))

%o         for i in range(N+1)

%o     ])

%o def matrix(N):

%o     return m(N) + m(N).transpose() - eye(N+1)

%o [ matrix(i).det() for i in range(20)]

%o # Gilles Castel, Sep 25 2017

%K sign

%O 0,4

%A _Alexander Farrugia_, Sep 25 2017