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A213268
Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.
4
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).
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
KEYWORD
nonn,frac,tabl
AUTHOR
Paul Curtz, Jun 08 2012
STATUS
approved