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Inverse of number triangle A(n,k) = 1/n! if k <= n <= 2k, 0 otherwise.
1

%I #5 Sep 29 2018 18:45:43

%S 1,0,1,0,-1,2,0,1,-2,6,0,0,0,-6,24,0,-1,2,0,-24,120,0,0,0,0,0,-120,

%T 720,0,1,-2,6,0,0,-720,5040,0,0,0,0,0,0,0,-5040,40320,0,0,0,-6,24,0,0,

%U 0,-40320,362880,0,0,0,0,0,0,0,0,0

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

%C It is conjectured that this is an integer triangle. Row sums are A127756.

%e Triangle begins

%e 1;

%e 0, 1;

%e 0, -1, 2;

%e 0, 1, -2, 6;

%e 0, 0, 0, -6, 24;

%e 0, -1, 2, 0, -24, 120;

%e 0, 0, 0, 0, 0, -120, 720;

%e 0, 1, -2, 6, 0, 0, -720, 5040;

%e 0, 0, 0, 0, 0, 0, 0, -5040, 40320;

%e 0, 0, 0, -6, 24, 0, 0, 0, -40320, 362880;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, -362880, 3628800;

%e 0, -1, 2, 0, -24, 120, 0, 0, 0, 0, -3628800, 39916800;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -39916800, 479001600;

%e Inverse of triangle begins

%e 1;

%e 0, 1;

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

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

%e 0, 0, 1/24, 1/24, 1/24;

%e 0, 0, 0, 1/120, 1/120, 1/120;

%e 0, 0, 0, 1/720, 1/720, 1/720, 1/720;

%e 0, 0, 0, 0, 1/5040, 1/5040, 1/5040, 1/5040;

%e 0, 0, 0, 0, 1/40320, 1/40320, 1/40320, 1/40320, 1/40320;

%e 0, 0, 0, 0, 0, 1/362880, 1/362880, 1/362880, 1/362880, 1/362880;

%e 0, 0, 0, 0, 0, 1/3628800, 1/3628800, 1/3628800, 1/3628800, 1/3628800, 1/3628800;

%K sign,tabl

%O 0,6

%A _Paul Barry_, Jan 28 2007