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Inverse of number triangle A(n,k) = 1/floor((n+2)/2) if k <= n <= 2k, 0 otherwise.
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%I #16 Oct 09 2022 21:16:05

%S 1,0,1,0,-1,2,0,1,-2,2,0,0,0,-2,3,0,-1,2,0,-3,3,0,0,0,0,0,-3,4,0,1,-2,

%T 2,0,0,-4,4,0,0,0,0,0,0,0,-4,5,0,0,0,-2,3,0,0,0,-5,5,0,0,0,0,0,0,0,0,

%U 0,-5,6

%N Inverse of number triangle A(n,k) = 1/floor((n+2)/2) if k <= n <= 2k, 0 otherwise.

%C It is conjectured that the triangle is an integer triangle. The triangle and its inverse both appear to have row sums equal to the all 1's sequence.

%C The triangle is equivalent to the lower semi-matrix R = e_{1,1} + Sum_{i>=2} Sum_{p>=0} ( e_{2^p i, i} ceiling(i/2) - e_{2^p (i+1), i} ceiling(i/2) ) , where e_{i,j} is the matrix unit. The conjecture above is true, deduced from the formula of the matrix. - _FUNG Cheok Yin_, Sep 12 2022

%H FUNG Cheok Yin, <a href="/A127793/a127793_1.txt">the triangle with the first 61 rows</a>

%e Triangle begins

%e 1;

%e 0, 1;

%e 0, -1, 2;

%e 0, 1, -2, 2;

%e 0, 0, 0, -2, 3;

%e 0, -1, 2, 0, -3, 3;

%e 0, 0, 0, 0, 0, -3, 4;

%e 0, 1, -2, 2, 0, 0, -4, 4;

%e 0, 0, 0, 0, 0, 0, 0, -4, 5;

%e 0, 0, 0, -2, 3, 0, 0, 0, -5, 5;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 6;

%e 0, -1, 2, 0, -3, 3, 0, 0, 0, 0, -6, 6;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 7;

%e Inverse of the triangle begins

%e 1;

%e 0, 1;

%e 0, 1/2, 1/2;

%e 0, 0, 1/2, 1/2;

%e 0, 0, 1/3, 1/3, 1/3;

%e 0, 0, 0, 1/3, 1/3, 1/3;

%e 0, 0, 0, 1/4, 1/4, 1/4, 1/4;

%e 0, 0, 0, 0, 1/4, 1/4, 1/4, 1/4;

%e 0, 0, 0, 0, 1/5, 1/5, 1/5, 1/5, 1/5;

%e 0, 0, 0, 0, 0, 1/5, 1/5, 1/5, 1/5, 1/5;

%e 0, 0, 0, 0, 0, 1/6, 1/6, 1/6, 1/6, 1/6, 1/6;

%t rows = 11;

%t A[n_, k_] := If[k <= n, If[n <= 2 k, 1/Floor[(n+2)/2] , 0], 0];

%t T = Table[A[n, k], {n, 0, rows-1}, {k, 0, rows-1}] // Inverse;

%t Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Stefano Spezia_, Sep 30 2018 *)

%K sign,tabl

%O 0,6

%A _Paul Barry_, Jan 29 2007