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A104984 Matrix inverse of triangle A104980.

%I A104984 #10 Jun 08 2021 14:58:01
%S A104984 1,-1,1,-1,-2,1,-3,-1,-3,1,-13,-3,-1,-4,1,-71,-13,-3,-1,-5,1,-461,-71,
%T A104984 -13,-3,-1,-6,1,-3447,-461,-71,-13,-3,-1,-7,1,-29093,-3447,-461,-71,
%U A104984 -13,-3,-1,-8,1,-273343,-29093,-3447,-461,-71,-13,-3,-1,-9,1,-2829325,-273343,-29093,-3447,-461,-71,-13,-3,-1,-10,1
%N A104984 Matrix inverse of triangle A104980.
%C A104984 Inverse matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.
%H A104984 G. C. Greubel, <a href="/A104984/b104984.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A104984 T(n, n) = 1, T(n+1, n) = -(n+1) for n >= 0; otherwise T(n, k) = T(n-k, 0) = -A003319(n-k-1) for n > k+1 and k >= 0.
%F A104984 Sum_{k=0..n} T(n, k) = A104985(n). - _G. C. Greubel_, Jun 07 2021
%e A104984 Triangle begins:
%e A104984        1;
%e A104984       -1,     1;
%e A104984       -1,    -2,    1;
%e A104984       -3,    -1,   -3,   1;
%e A104984      -13,    -3,   -1,  -4,   1;
%e A104984      -71,   -13,   -3,  -1,  -5,  1;
%e A104984     -461,   -71,  -13,  -3,  -1, -6,  1;
%e A104984    -3447,  -461,  -71, -13,  -3, -1, -7,  1;
%e A104984   -29093, -3447, -461, -71, -13, -3, -1, -8, 1; ...
%t A104984 A003319[n_]:= A003319[n]= If[n==0, 0, n! - Sum[j!*A003319[n-j], {j,n-1}]];
%t A104984 T[n_, k_]:= If[k==n, 1, If[k==n-1, -n, -A003319[n-k]]];
%t A104984 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 07 2021 *)
%o A104984 (PARI) T(n,k)=if(n==k,1,if(n==k+1,-n,-(n-k)!-sum(i=1,n-k-1,i!*T(n-k-i,0))));
%o A104984 (Sage)
%o A104984 @CachedFunction
%o A104984 def T(n,k):
%o A104984     if (k==n): return 1
%o A104984     elif (k==n-1): return -n
%o A104984     else: return -factorial(n-k) - sum( factorial(j)*T(n-k-j, 0) for j in (1..n-k-1) )
%o A104984 [[T(n,k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jun 07 2021
%Y A104984 Cf. A104980, A104985 (row sums).
%K A104984 sign,tabl
%O A104984 0,5
%A A104984 _Paul D. Hanna_, Apr 10 2005