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A057728 A triangular table of decreasing powers of two (with first column all ones). 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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

LINKS

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

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)

EXAMPLE

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 of a(n) 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)

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 *)

PROG

(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

(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 */

CROSSREFS

Cf. A000079, A004736, A023758 and A000337.

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

Sequence in context: A141020 A152568 A155038 * A176463 A098050 A278984

Adjacent sequences:  A057725 A057726 A057727 * A057729 A057730 A057731

KEYWORD

base,easy,nonn,tabl

AUTHOR

Alford Arnold, Oct 29 2000

EXTENSIONS

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

STATUS

approved

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Last modified December 9 09:20 EST 2019. Contains 329877 sequences. (Running on oeis4.)