Symmetric Pascal matrices modulo p

T =  1 1 1 1 2 1 1 3 3 1 .. . . .  = exp  0 1 0 0 2 0 0 3 0 . . .  with coefficients ti,j = (i j ) . This shows that det(P (n)) = 1 and that P (n) is positive definite for all n ∈ N. It implies furthermore that the characteristic polynomial det(tI(n)−P (n)) = ∑ k=0 αkt k (where I(n) denotes the identity matrix of order n) of P (n) has only positive real roots. The inverse P (n)−1 of P (n) is given by