A352988 Matrix inverse of triangle A352650.
1, 0, 1, -1, -1, 1, 0, -2, -2, 1, 0, 0, -3, -3, 1, 0, 0, 0, -4, -4, 1, 0, 0, 0, 0, -5, -5, 1, 0, 0, 0, 0, 0, -6, -6, 1, 0, 0, 0, 0, 0, 0, -7, -7, 1, 0, 0, 0, 0, 0, 0, 0, -8, -8, 1, 0, 0, 0, 0, 0, 0, 0, 0, -9, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, -10, 1
Offset: 0
Examples
The triangle T(n,k) for 0 <= k <= n starts: n\k : 0 1 2 3 4 5 6 7 8 9 ====================================================== 0 : 1 1 : 0 1 2 : -1 -1 1 3 : 0 -2 -2 1 4 : 0 0 -3 -3 1 5 : 0 0 0 -4 -4 1 6 : 0 0 0 0 -5 -5 1 7 : 0 0 0 0 0 -6 -6 1 8 : 0 0 0 0 0 0 -7 -7 1 9 : 0 0 0 0 0 0 0 -8 -8 1 etc.
Formula
T(n,n) = 1 for n >= 0, and T(n,n-1) = 1 - n for n > 0, and T(n,n-2) = 1 - n for n > 1, and T(n,k) = 0 if n < 0 or k < 0 or n < k or n > k+2.
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = (1 + t) * (1 - (1 + x) * t) / (1 - x * t)^2.
Alt. row sums equal (-1)^n for n >= 0.