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A127749 Inverse of number triangle A(n,k) = 1/(2n+1) if k <= n <= 2k, 0 otherwise.

%I A127749 #13 Apr 21 2021 11:33:18
%S A127749 1,0,3,0,-3,5,0,3,-5,7,0,0,0,-7,9,0,-3,5,0,-9,11,0,0,0,0,0,-11,13,0,3,
%T A127749 -5,7,0,0,-13,15,0,0,0,0,0,0,0,-15,17,0,0,0,-7,9,0,0,0,-17,19,0,0,0,0,
%U A127749 0,0,0,0,0,-19,21,0,-3,5
%N A127749 Inverse of number triangle A(n,k) = 1/(2n+1) if k <= n <= 2k, 0 otherwise.
%C A127749 Conjectures: row sums modulo 2 are the Fredholm-Rueppel sequence A036987; row sums of triangle modulo 2 are A111982. Row sums are A127750.
%C A127749 The first conjecture is equivalent to the row sums conjecture in A111967. - _R. J. Mathar_, Apr 21 2021
%F A127749 T(n,k) = (2*k+1)*A111967(n,k). - _R. J. Mathar_, Apr 21 2021
%e A127749 Triangle begins
%e A127749   1;
%e A127749   0,  3;
%e A127749   0, -3,  5;
%e A127749   0,  3, -5,  7;
%e A127749   0,  0,  0, -7,  9;
%e A127749   0, -3,  5,  0, -9,  11;
%e A127749   0,  0,  0,  0,  0, -11,  13;
%e A127749   0,  3, -5,  7,  0,   0, -13,  15;
%e A127749   0,  0,  0,  0,  0,   0,   0, -15,  17;
%e A127749   0,  0,  0, -7,  9,   0,   0,   0, -17,  19;
%e A127749   0,  0,  0,  0,  0,   0,   0,   0,   0, -19,  21;
%e A127749   0, -3,  5,  0, -9,  11,   0,   0,   0,   0, -21,  23;
%e A127749   0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0, -23, 25;
%e A127749 Inverse of triangle
%e A127749   1;
%e A127749   0, 1/3;
%e A127749   0, 1/5, 1/5;
%e A127749   0,  0,  1/7, 1/7;
%e A127749   0,  0,  1/9, 1/9,  1/9;
%e A127749   0,  0,   0,  1/11, 1/11, 1/11;
%e A127749   0,  0,   0,  1/13, 1/13, 1/13, 1/13;
%e A127749   0,  0,   0,   0,   1/15, 1/15, 1/15, 1/15;
%e A127749   0,  0,   0,   0,   1/17, 1/17, 1/17, 1/17, 1/17;
%e A127749   0,  0,   0,   0,    0,   1/19, 1/19, 1/19, 1/19, 1/19;
%e A127749   0,  0,   0,   0,    0,   1/21, 1/21, 1/21, 1/21, 1/21, 1/21;
%p A127749 A127749 := proc(n,k)
%p A127749     option remember ;
%p A127749     if k > n then
%p A127749         0 ;
%p A127749     elif k = n then
%p A127749         2*n+1 ;
%p A127749     else
%p A127749         -(2*k+1)*add( procname(n,i)/(2*i+1),i=k+1..min(n,2*k)) ;
%p A127749     end if;
%p A127749 end proc:
%p A127749 seq(seq( A127749(n,k),k=0..n),n=0..20) ; # _R. J. Mathar_, Feb 09 2021
%t A127749 nmax = 10;
%t A127749 A[n_, k_] := If[k <= n <= 2k, 1/(2n+1), 0];
%t A127749 invA = Inverse[Table[A[n, k], {n, 0, nmax}, {k, 0, nmax}]];
%t A127749 T[n_, k_] := invA[[n+1, k+1]];
%t A127749 Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 05 2020 *)
%Y A127749 Cf. A111967.
%K A127749 sign,tabl
%O A127749 0,3
%A A127749 _Paul Barry_, Jan 28 2007