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Matrix inverse of triangle A104980.
4

%I #10 Jun 08 2021 14:58:01

%S 1,-1,1,-1,-2,1,-3,-1,-3,1,-13,-3,-1,-4,1,-71,-13,-3,-1,-5,1,-461,-71,

%T -13,-3,-1,-6,1,-3447,-461,-71,-13,-3,-1,-7,1,-29093,-3447,-461,-71,

%U -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 Matrix inverse of triangle A104980.

%C Inverse matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.

%H G. C. Greubel, <a href="/A104984/b104984.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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 Sum_{k=0..n} T(n, k) = A104985(n). - _G. C. Greubel_, Jun 07 2021

%e Triangle begins:

%e 1;

%e -1, 1;

%e -1, -2, 1;

%e -3, -1, -3, 1;

%e -13, -3, -1, -4, 1;

%e -71, -13, -3, -1, -5, 1;

%e -461, -71, -13, -3, -1, -6, 1;

%e -3447, -461, -71, -13, -3, -1, -7, 1;

%e -29093, -3447, -461, -71, -13, -3, -1, -8, 1; ...

%t A003319[n_]:= A003319[n]= If[n==0, 0, n! - Sum[j!*A003319[n-j], {j,n-1}]];

%t T[n_, k_]:= If[k==n, 1, If[k==n-1, -n, -A003319[n-k]]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 07 2021 *)

%o (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 (Sage)

%o @CachedFunction

%o def T(n,k):

%o if (k==n): return 1

%o elif (k==n-1): return -n

%o else: return -factorial(n-k) - sum( factorial(j)*T(n-k-j, 0) for j in (1..n-k-1) )

%o [[T(n,k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jun 07 2021

%Y Cf. A104980, A104985 (row sums).

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Apr 10 2005