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

0,5

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).

Links

Table of n, a(n) for n=0..77.

Example

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.

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 *)

Crossrefs

Sequence in context: A266349 A219094 A362824 * A291118 A245184 A284414

Adjacent sequences: A213265 A213266 A213267 * A213269 A213270 A213271

Keyword

nonn,frac,tabl

Author

Paul Curtz, Jun 08 2012

Status

approved