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%I #22 Apr 07 2025 13:46:28
%S 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,
%T 21,-6,1,-120,360,-420,260,-105,31,-7,1,720,-2160,2520,-1560,630,-186,
%U 43,-8,1,-5040,15120,-17640,10920,-4410,1302,-301,57,-9,1
%N Matrix inverse of the rascal triangle (A077028), read across rows.
%C The columns of this triangle are related to factorial numbers (A000142).
%C 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).
%H Werner Schulte, <a href="/A338291/a338291.pdf">Proof of the Generalized Formula</a>
%F 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.
%F T(n,k) = (2-n) * T(n-1,k) for 0 <= k < n-2.
%F T(n,k) = T(k+2,k) * (-1)^(n-k) * (n-2)! / k! for 0 <= k <= n-2.
%F Row sums are A000007(n) for n >= 0.
%e The triangle T(n,k) for 0 <= k <= n starts:
%e n\k : 0 1 2 3 4 5 6 7 8 9
%e ================================================================
%e 0 : 1
%e 1 : -1 1
%e 2 : 1 -2 1
%e 3 : -1 3 -3 1
%e 4 : 2 -6 7 -4 1
%e 5 : -6 18 -21 13 -5 1
%e 6 : 24 -72 84 -52 21 -6 1
%e 7 : -120 360 -420 260 -105 31 -7 1
%e 8 : 720 -2160 2520 -1560 630 -186 43 -8 1
%e 9 : -5040 15120 -17640 10920 -4410 1302 -301 57 -9 1
%e etc.
%o (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!,", ")))))
%o (PARI) 1/matrix(10, 10, n, k, n--; k--; if (n>=k, k*(n-k) + 1)) \\ _Michel Marcus_, Nov 11 2020
%Y Cf. A000007, A000142, A077028.
%K sign,easy,tabl
%O 0,5
%A _Werner Schulte_, Oct 20 2020