A127803 - OEIS

%I #17 Feb 12 2024 08:17:17

%S 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 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 -511,1023,0,0,0,0,0,0,0,0,0,-1023,2047

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

%C Row sums are A127804.

%e Triangle begins

%e   1;

%e   0,  3;

%e   0, -3,  7;

%e   0,  3, -7,  15;

%e   0,  0,  0, -15,  31;

%e   0, -3,  7,   0, -31,  63;

%e   0,  0,  0,   0,   0, -63,  127;

%e   0,  3, -7,  15,   0,   0, -127,  255;

%e   0,  0,  0,   0,   0,   0,    0, -255,  511;

%e   0,  0,  0, -15,  31,   0,    0,    0, -511,  1023;

%e   0,  0,  0,   0,   0,   0,    0,    0,    0, -1023, 2047;

%e   ...

%e Inverse of

%e   1;

%e   0, 1/3;

%e   0, 1/7, 1/7;

%e   0,  0,  1/15, 1/15;

%e   0,  0,  1/31, 1/31,  1/31;

%e   0,  0,   0,   1/63,  1/63,  1/63;

%e   0,  0,   0,   1/127, 1/127, 1/127,  1/127;

%e   0,  0,   0,    0,    1/255, 1/255,  1/255,  1/255;

%e   0,  0,   0,    0,    1/511, 1/511,  1/511,  1/511,  1/511;

%e   0,  0,   0,    0,     0,    1/1023, 1/1023, 1/1023, 1/1023, 1/1023;

%e   0,  0,   0,    0,     0,    1/2047, 1/2047, 1/2047, 1/2047, 1/2047, 1/2047;

%e   ...

%p A127803 := proc(n,k)

%p     A := Matrix(n+1,n+1) ;

%p     for r from 0 to n do

%p     for c from 0 to n do

%p         if c <= r and r <= 2*c then

%p             A[r+1,c+1] := 1/(2*2^r-1) ;

%p         else

%p             A[r+1,c+1] := 0 ;

%p         end if;

%p     end do:

%p     end do:

%p     Ainv := LinearAlgebra[MatrixInverse](A) ;

%p     Ainv[n+1,k+1] ;

%p end proc:

%p seq(seq( A127803(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Feb 12 2024

%t rows = 11;

%t A[n_, k_] := If[k <= n, If[n <= 2 k, 1/(2*2^n - 1), 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 (* _Jean-François Alcover_, Jul 03 2018 *)

%o (PARI) B(n,k) = if(k<=n,if(n<=2*k,1/(2*2^n-1),0),0);

%o 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 Cf. A127804.

%K sign,tabl

%O 0,3

%A _Paul Barry_, Jan 29 2007