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%I #32 Jan 15 2022 14:55:00
%S 1,1,1,1,2,1,1,4,2,1,1,8,4,2,1,1,16,8,4,2,1,1,32,16,8,4,2,1,1,64,32,
%T 16,8,4,2,1,1,128,64,32,16,8,4,2,1,1,256,128,64,32,16,8,4,2,1,1,512,
%U 256,128,64,32,16,8,4,2,1,1,1024,512,256,128,64,32,16,8,4,2,1,1,2048,1024,512,256,128,64,32,16,8,4,2,1
%N A triangular table of decreasing powers of two (with first column all ones).
%C First differences of sequence A023758.
%C A023758 is the sequence of partial sums of a(n) with row sums A000337.
%C 2^A004736(n) is a sequence closely related to a(n).
%C T(n,k) is the number of length n binary words having an odd number of 0's with exactly k 1's following the last 0, n >= 1, 0 <= k <= n - 1. - _Geoffrey Critzer_, Jan 28 2014
%H Reinhard Zumkeller, <a href="/A057728/b057728.txt">Rows n = 1..100 of table, flattened</a>
%F G.f.: (x - x^2)/((1 - 2*x)*(1 - y*x)). - _Geoffrey Critzer_, Jan 28 2014 [This produces the triangle shown by _Mats Granvik_ in example section. - _Franck Maminirina Ramaharo_, Jan 09 2019]
%F From _Franck Maminirina Ramaharo_, Jan 09 2019: (Start)
%F G.f.: x*(1 - 2*x + y*x^2)/((1 - x)*(1 - 2*x)*(1 - x*y)).
%F E.g.f.: (exp(2*x)*y - 2*exp(x*y))/(4 - 2*y) + exp(x) - 1/2. (End)
%e Triangle starts:
%e 1,
%e 1, 1,
%e 1, 2, 1,
%e 1, 4, 2, 1,
%e 1, 8, 4, 2, 1,
%e 1, 16, 8, 4, 2, 1,
%e 1, 32, 16, 8, 4, 2, 1,
%e 1, 64, 32, 16, 8, 4, 2, 1,
%e 1, 128, 64, 32, 16, 8, 4, 2, 1,
%e 1, 256, 128, 64, 32, 16, 8, 4, 2, 1,
%e 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1,
%e 1, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1,
%e 1, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1,
%e ... - _Joerg Arndt_, May 04 2014
%e When viewed as a triangular array, row 8 of A023758 is 128 192 224 240 248 252 254 255 so row 8 here is 1 64 32 16 8 4 2 1
%e From _Mats Granvik_, Jan 19 2009: (Start)
%e Except for the first term the table can also be formatted as:
%e 1,
%e 1, 1,
%e 2, 1, 1,
%e 4, 2, 1, 1,
%e 8, 4, 2, 1, 1,
%e 16, 8, 4, 2, 1, 1,
%e ...
%e (End)
%t nn=10;Map[Select[#,#>0&]&,CoefficientList[Series[(x-x^2)/(1-2x)/(1-y x),{x,0,nn}],{x,y}]]//Grid (* _Geoffrey Critzer_, Jan 28 2014 *)
%t Module[{nn=12,ts},ts=2^Range[0,nn];Table[Join[{1},Reverse[Take[ts,n]]],{n,0,nn}]]//Flatten (* _Harvey P. Dale_, Jan 15 2022 *)
%o (Haskell)
%o a057728 n k = a057728_tabl !! (n-1) !! (k-1)
%o a057728_row n = a057728_tabl !! (n-1)
%o a057728_tabl = iterate
%o (\row -> zipWith (+) (row ++ [0]) ([0] ++ tail row ++ [1])) [1]
%o -- _Reinhard Zumkeller_, Aug 08 2013
%o (Maxima)
%o T(n, k) := if k = 0 then 1 else 2^(n - k - 1)$
%o create_list(T(n, k), n, 0, 12, k, 0, n - 1); /* _Franck Maminirina Ramaharo_, Jan 09 2019 */
%Y Cf. A000079, A004736, A023758 and A000337.
%Y Cf. A155038 (essentially the same as this sequence). [_Mats Granvik_, Jan 19 2009]
%K base,easy,nonn,tabl
%O 1,5
%A _Alford Arnold_, Oct 29 2000
%E More terms from Larry Reeves (larryr(AT)acm.org), Oct 30 2000
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