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%I #23 Jun 09 2016 11:54:50
%S 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,
%T 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,
%U 8,32,4,128,128,512,256,1,1,4,4,16,2,64,64,256,128,1024,1024
%N Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.
%C 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.
%C Here, where the first row is the nonnegative integers, the array is
%C 0 1 2 3 4 5 6 7 8 =A001477(n)
%C 1 3/2 2 5/2 3 7/2 4 9/2 5 =A026741(n+2)/A000034(n)
%C 1 5/4 3/2 7/4 2 9/4 5/2 11/4 3 =A060819(n+4)/A176895(n)
%C 3/4 7/8 1 9/8 5/4 11/8 3/2 13/8 7/4 =A106609(n+6)/A205383(n+6)
%C 1/2 9/16 5/8 11/16 3/4 13/16 7/8 15/16 1 =A106617(n+8)/TBD
%C 5/16 11/32 3/8 13/32 7/16 15/32 1/2 17/32 9/16
%C 3/16 13/64 7/32 15/64 1/4 17/64 9/32 19/64 5/16
%C 7/64 15/128 1/8 17/128 9/64 19/128 5/32 21/128 11/64
%C 1/16 17/256 9/128 19/256 5/64 21/256 11/128 23/256 3/32.
%C The first column contains 0, followed by fractions A000265/A084623, that is Oresme numbers n/2^n multiplied by 2 (see A209308).
%e The array of denominators starts:
%e 1 1 1 1 1 1 1 1 1 1 1 ...
%e 1 2 1 2 1 2 1 2 1 2 1 ...
%e 1 4 2 4 1 4 2 4 1 4 2 ...
%e 4 8 1 8 4 8 2 8 4 8 1 ...
%e 2 16 8 16 4 16 8 16 1 16 8 ...
%e 16 32 8 32 16 32 2 32 16 32 8 ...
%e 16 64 32 64 4 64 32 64 16 64 32 ...
%e 64 128 8 128 64 128 32 128 64 128 16 ...
%e 16 256 128 256 64 256 128 256 32 256 128 ...
%e 256 512 128 512 256 512 64 512 256 512 128 ...
%e All entries are powers of 2.
%p A213268frac := proc(n,k)
%p if n = 0 then
%p return k ;
%p else
%p return procname(n-1,k+1)-procname(n-1,k)/2 ;
%p end if;
%p end proc:
%p A213268 := proc(n,k)
%p denom(A213268frac(n,k)) ;
%p end proc: # _R. J. Mathar_, Jun 30 2012
%t 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 *)
%K nonn,frac,tabl
%O 0,5
%A _Paul Curtz_, Jun 08 2012
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