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A127803 Inverse of number triangle A(n,k) = 1/(2*2^n-1) if k <= n <= 2k, 0 otherwise.

%I A127803 #22 Jun 11 2025 01:02:45
%S A127803 1,0,3,0,-3,7,0,3,-7,15,0,0,0,-15,31,0,-3,7,0,-31,63,0,0,0,0,0,-63,
%T A127803 127,0,3,-7,15,0,0,-127,255,0,0,0,0,0,0,0,-255,511,0,0,0,-15,31,0,0,0,
%U A127803 -511,1023,0,0,0,0,0,0,0,0,0,-1023,2047
%N A127803 Inverse of number triangle A(n,k) = 1/(2*2^n-1) if k <= n <= 2k, 0 otherwise.
%C A127803 Row sums are A127804.
%H A127803 Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Sequence_A127804_from_signed_triangle.svg">Illustration of first 32 rows</a>
%e A127803 Triangle begins
%e A127803   1;
%e A127803   0,  3;
%e A127803   0, -3,  7;
%e A127803   0,  3, -7,  15;
%e A127803   0,  0,  0, -15,  31;
%e A127803   0, -3,  7,   0, -31,  63;
%e A127803   0,  0,  0,   0,   0, -63,  127;
%e A127803   0,  3, -7,  15,   0,   0, -127,  255;
%e A127803   0,  0,  0,   0,   0,   0,    0, -255,  511;
%e A127803   0,  0,  0, -15,  31,   0,    0,    0, -511,  1023;
%e A127803   0,  0,  0,   0,   0,   0,    0,    0,    0, -1023, 2047;
%e A127803   ...
%e A127803 Inverse of
%e A127803   1;
%e A127803   0, 1/3;
%e A127803   0, 1/7, 1/7;
%e A127803   0,  0,  1/15, 1/15;
%e A127803   0,  0,  1/31, 1/31,  1/31;
%e A127803   0,  0,   0,   1/63,  1/63,  1/63;
%e A127803   0,  0,   0,   1/127, 1/127, 1/127,  1/127;
%e A127803   0,  0,   0,    0,    1/255, 1/255,  1/255,  1/255;
%e A127803   0,  0,   0,    0,    1/511, 1/511,  1/511,  1/511,  1/511;
%e A127803   0,  0,   0,    0,     0,    1/1023, 1/1023, 1/1023, 1/1023, 1/1023;
%e A127803   0,  0,   0,    0,     0,    1/2047, 1/2047, 1/2047, 1/2047, 1/2047, 1/2047;
%e A127803   ...
%p A127803 A127803 := proc(n,k)
%p A127803     A := Matrix(n+1,n+1) ;
%p A127803     for r from 0 to n do
%p A127803     for c from 0 to n do
%p A127803         if c <= r and r <= 2*c then
%p A127803             A[r+1,c+1] := 1/(2*2^r-1) ;
%p A127803         else
%p A127803             A[r+1,c+1] := 0 ;
%p A127803         end if;
%p A127803     end do:
%p A127803     end do:
%p A127803     Ainv := LinearAlgebra[MatrixInverse](A) ;
%p A127803     Ainv[n+1,k+1] ;
%p A127803 end proc:
%p A127803 seq(seq( A127803(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Feb 12 2024
%t A127803 rows = 11;
%t A127803 A[n_, k_] := If[k <= n, If[n <= 2 k, 1/(2*2^n - 1), 0], 0];
%t A127803 T = Table[A[n, k], {n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
%t A127803 Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 03 2018 *)
%o A127803 (PARI) B(n,k) = if(k<=n,if(n<=2*k,1/(2*2^n-1),0),0);
%o A127803 lista(nn) = {my(m = matrix(nn, nn, n, k, B(n-1,k-1))^(-1)); for (n=1, nn, for (k=1, n, print1(m[n,k], ", ");); print(););} \\ _Michel Marcus_, Jul 03 2018
%Y A127803 Cf. A127804.
%K A127803 sign,tabl
%O A127803 0,3
%A A127803 _Paul Barry_, Jan 29 2007