A057728 - OEIS

%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