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Inverse of a double factorial related triangle.
5

%I #6 Feb 17 2021 19:31:17

%S 1,-1,1,0,-3,1,0,0,-5,1,0,0,0,-7,1,0,0,0,0,-9,1,0,0,0,0,0,-11,1,0,0,0,

%T 0,0,0,-13,1,0,0,0,0,0,0,0,-15,1,0,0,0,0,0,0,0,0,-17,1,0,0,0,0,0,0,0,

%U 0,0,-19,1,0,0,0,0,0,0,0,0,0,0,-21,1,0,0,0,0,0,0,0,0,0,0,0,-23,1

%N Inverse of a double factorial related triangle.

%C Inverse of A112292. Similar results can be obtained for higher factorials.

%H G. C. Greubel, <a href="/A112295/b112295.txt">Rows n = 0..50 of the triangle, flattened</a>

%F From _G. C. Greubel_, Feb 17 2021: (Start)

%F T(n, k) = 1 - 2*n if k = n-1 otherwise 0, with T(n, n) = 1.

%F Sum_{k=0..n} T(n, k) = 1 - 2*n - [n=0]. (End)

%e Triangle begins

%e 1;

%e -1, 1;

%e 0, -3, 1;

%e 0, 0, -5, 1;

%e 0, 0, 0, -7, 1;

%e 0, 0, 0, 0, -9, 1;

%e 0, 0, 0, 0, 0, -11, 1;

%t T[n_, k_]:= If[k==n, 1, If[k==n-1, 1-2*n, 0]];

%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 17 2021 *)

%o (Sage)

%o def A112295(n,k): return 1 if k==n else 1-2*n if k==n-1 else 0

%o flatten([[A112295(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Feb 17 2021

%o (Magma)

%o A112295:= func< n,k | k eq n select 1 else k eq n-1 select 1-2*n else 0 >;

%o [A112295(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Feb 17 2021

%Y Cf. A094587, A134081.

%K sign,tabl

%O 0,5

%A _Paul Barry_, Sep 01 2005