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A057728
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A triangular table of decreasing powers of two (with first column all ones).
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8
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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
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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EXAMPLE
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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,
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
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)
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MATHEMATICA
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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 *)
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PROG
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(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]
(Maxima)
T(n, k) := if k = 0 then 1 else 2^(n - k - 1)$
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Oct 30 2000
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STATUS
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approved
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