A057728 A triangular table of decreasing powers of two (with first column all ones).

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, 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, 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

Offset: 1

Author

Alford Arnold, Oct 29 2000

Keywords

• base
• easy
• nonn
• tabl

Comments

First differences of sequence A023758.

A023758 is the sequence of partial sums of a(n) with row sums A000337.

2^A004736(n) is a sequence closely related to a(n).

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

Examples

			Triangle starts:
  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,  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,  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,
  ... - _Joerg Arndt_, May 04 2014
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
From _Mats Granvik_, Jan 19 2009: (Start)
Except for the first term the table can also be formatted as:
   1,
   1, 1,
   2, 1, 1,
   4, 2, 1, 1,
   8, 4, 2, 1, 1,
  16, 8, 4, 2, 1, 1,
  ...
(End)
		

Links

• Reinhard Zumkeller, Rows n = 1..100 of table, flattened

Crossrefs

Cf. A000079, A004736, A023758 and A000337.

Cf. A155038 (essentially the same as this sequence). [Mats Granvik, Jan 19 2009]

Programs

Haskell

a057728 n k = a057728_tabl !! (n-1) !! (k-1)
a057728_row n = a057728_tabl !! (n-1)
a057728_tabl = iterate
   (\row -> zipWith (+) (row ++ [0]) ([0] ++ tail row ++ [1])) [1]
-- Reinhard Zumkeller, Aug 08 2013

Mathematica

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 *)
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 *)

Maxima

T(n, k) := if k = 0 then 1 else  2^(n - k - 1)$
create_list(T(n, k), n, 0, 12, k, 0, n - 1); /* Franck Maminirina Ramaharo, Jan 09 2019 */

Formula

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]

From Franck Maminirina Ramaharo, Jan 09 2019: (Start)

G.f.: x*(1 - 2*x + y*x^2)/((1 - x)*(1 - 2*x)*(1 - x*y)).

E.g.f.: (exp(2*x)*y - 2*exp(x*y))/(4 - 2*y) + exp(x) - 1/2. (End)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 30 2000