A213268 Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.

1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 2, 2, 8, 2, 1, 1, 4, 1, 16, 16, 1, 2, 1, 8, 8, 32, 16, 1, 1, 4, 4, 16, 8, 64, 64, 1, 2, 2, 8, 4, 32, 32, 128, 16, 1, 1, 4, 2, 16, 16, 64, 8, 256, 256, 1, 2, 1, 8, 8, 32, 4, 128, 128, 512, 256, 1, 1, 4, 4, 16, 2, 64, 64, 256, 128, 1024, 1024

Offset: 0

Author

Paul Curtz, Jun 08 2012

Keywords

• nonn
• frac
• tabl

Comments

Starting from any sequence a(k) in the first row, define the array T(n,k) of the inverse semi-binomial transform by T(0,k) = a(k), T(n,k) = T(n-1,k+1) -T(n-1,k)/2, n>=1.

Here, where the first row is the nonnegative integers, the array is

0 1 2 3 4 5 6 7 8 =A001477(n)

1 3/2 2 5/2 3 7/2 4 9/2 5 =A026741(n+2)/A000034(n)

1 5/4 3/2 7/4 2 9/4 5/2 11/4 3 =A060819(n+4)/A176895(n)

3/4 7/8 1 9/8 5/4 11/8 3/2 13/8 7/4 =A106609(n+6)/A205383(n+6)

1/2 9/16 5/8 11/16 3/4 13/16 7/8 15/16 1 =A106617(n+8)/TBD

5/16 11/32 3/8 13/32 7/16 15/32 1/2 17/32 9/16

3/16 13/64 7/32 15/64 1/4 17/64 9/32 19/64 5/16

7/64 15/128 1/8 17/128 9/64 19/128 5/32 21/128 11/64

1/16 17/256 9/128 19/256 5/64 21/256 11/128 23/256 3/32.

The first column contains 0, followed by fractions A000265/A084623, that is Oresme numbers n/2^n multiplied by 2 (see A209308).

Examples

			The array of denominators starts:
  1   1   1   1   1   1   1   1   1   1   1 ...
  1   2   1   2   1   2   1   2   1   2   1 ...
  1   4   2   4   1   4   2   4   1   4   2 ...
  4   8   1   8   4   8   2   8   4   8   1 ...
  2  16   8  16   4  16   8  16   1  16   8 ...
16  32   8  32  16  32   2  32  16  32   8 ...
16  64  32  64   4  64  32  64  16  64  32 ...
64 128   8 128  64 128  32 128  64 128  16 ...
16 256 128 256  64 256 128 256  32 256 128 ...
256 512 128 512 256 512  64 512 256 512 128 ...
All entries are powers of 2.
		

Programs

Maple

A213268frac := proc(n,k)
        if n = 0 then
                return k ;
        else
                return procname(n-1,k+1)-procname(n-1,k)/2 ;
        end if;
end proc:
A213268 := proc(n,k)
        denom(A213268frac(n,k)) ;
end proc: # R. J. Mathar, Jun 30 2012

Mathematica

T[0, k_] := k; T[n_, k_] := T[n, k] = T[n-1, k+1] - T[n-1, k]/2; Table[T[n-k, k] // Denominator, {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2014 *)