OFFSET
0,5
COMMENTS
The columns of this triangle are related to factorial numbers (A000142).
There is a family of triangles T(m;n,k) = 1 + m*k*(n-k) for some fixed integer m (for m >= 0 see A296180, Comments) and 0 <= k <= n. They satisfy the equation T(-m;n,k) = 2 - T(m;n,k). The corresponding matrices inverse M = T^(-1) are given by: M(m;n,n) = 1 for n >= 0, and M(m;n,n-1) = m*(1-n) - 1 for n > 0, and M(m;n,k) = (-1)^(n-k) * m * (m * k*(k+1) + 1) * Product_{i=k+1..n-2} (m*(i+1) - 1) for 0 <= k <= n-2. For special cases of the M(m;n,k) see A338817 (m=-1), and A167374 (m=0), and this triangle (m=1).
LINKS
Werner Schulte, Proof of the Generalized Formula
FORMULA
T(n,n) = 1 for n >= 0, and T(n,n-1) = -n for n > 0, and T(n,n-2) = n^2 - 3*n + 3 for n > 1, and T(n,k) = (-1)^(n-k) * (k^2 + k + 1) * (n-2)! / k! for 0 <= k <= n-2.
T(n,k) = (2-n) * T(n-1,k) for 0 <= k < n-2.
T(n,k) = T(k+2,k) * (-1)^(n-k) * (n-2)! / k! for 0 <= k <= n-2.
Row sums are A000007(n) for n >= 0.
EXAMPLE
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 : -1 1
2 : 1 -2 1
3 : -1 3 -3 1
4 : 2 -6 7 -4 1
5 : -6 18 -21 13 -5 1
6 : 24 -72 84 -52 21 -6 1
7 : -120 360 -420 260 -105 31 -7 1
8 : 720 -2160 2520 -1560 630 -186 43 -8 1
9 : -5040 15120 -17640 10920 -4410 1302 -301 57 -9 1
etc.
PROG
(PARI) for(n=0, 10, for(k=0, n, if(k==n, print(" 1"), if(k==n-1, print1(-n, ", "), print1((-1)^(n-k)*(k^2+k+1)*(n-2)!/k!, ", ")))))
(PARI) 1/matrix(10, 10, n, k, n--; k--; if (n>=k, k*(n-k) + 1)) \\ Michel Marcus, Nov 11 2020
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Oct 20 2020
STATUS
approved