A338291 Matrix inverse of the rascal triangle (A077028), read across rows.
1, -1, 1, 1, -2, 1, -1, 3, -3, 1, 2, -6, 7, -4, 1, -6, 18, -21, 13, -5, 1, 24, -72, 84, -52, 21, -6, 1, -120, 360, -420, 260, -105, 31, -7, 1, 720, -2160, 2520, -1560, 630, -186, 43, -8, 1, -5040, 15120, -17640, 10920, -4410, 1302, -301, 57, -9, 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 -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.
Links
- Werner Schulte, Proof of the Generalized Formula
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,", "),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
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.
Comments