A338817 Matrix inverse of triangle A176270, read by rows.
1, -1, 1, -1, 0, 1, -3, 1, 1, 1, -12, 4, 5, 2, 1, -60, 20, 25, 11, 3, 1, -360, 120, 150, 66, 19, 4, 1, -2520, 840, 1050, 462, 133, 29, 5, 1, -20160, 6720, 8400, 3696, 1064, 232, 41, 6, 1, -181440, 60480, 75600, 33264, 9576, 2088, 369, 55, 7, 1, -1814400, 604800, 756000, 332640, 95760, 20880, 3690, 550, 71, 8, 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 : -1 1 2 : -1 0 1 3 : -3 1 1 1 4 : -12 4 5 2 1 5 : -60 20 25 11 3 1 6 : -360 120 150 66 19 4 1 7 : -2520 840 1050 462 133 29 5 1 8 : -20160 6720 8400 3696 1064 232 41 6 1 9 : -181440 60480 75600 33264 9576 2088 369 55 7 1 etc.
Programs
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PARI
for(n=0,10,for(k=0,n,if(k==n,print(" 1"),if(k==n-1,print1(n-2,", "),print1((k^2+k-1)*n!/(k+2)!,", "))))) -
PARI
1/matrix(10, 10, n, k, n--; k--; if (n>=k, 1 + k*(k-n))) \\ Michel Marcus, Nov 11 2020
Formula
T(n,n) = 1 for n >= 0; T(n,n-1) = n - 2 for n > 0; T(n,n-2) = n^2 - 3*n + 1 for n > 1; T(n,k) = (k^2 + k - 1) * n! / (k+2)! for 0 <= k <= n-2.
T(n,k) = n * T(n-1,k) for 0 <= k < n-2.
T(n,k) = T(k+2,k) * n! / (k+2)! for 0 <= k <= n-2.
Row sums are A000007(n) for n >= 0.