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A209308 Denominators of the Akiyama-Tanigawa algorithm applied to 2^(-n), written by antidiagonals. 2
1, 2, 2, 1, 2, 4, 4, 4, 8, 8, 1, 4, 8, 4, 16, 2, 2, 1, 8, 32, 32, 1, 2, 4, 4, 16, 32, 64, 8, 8, 16, 16, 64, 64, 128, 128, 1, 8, 16, 8, 32, 64, 128, 32, 256, 2, 2, 8, 16, 64, 64, 128, 64, 512, 512, 1, 2, 4, 8, 32, 64, 128, 16, 128, 512, 1024 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

1/2^n and successive rows are

1,       1/2,   1/4,   1/8,  1/16,  1/32,   1/64, 1/128, 1/256,...

1/2,     1/2,   3/8,   1/4,  5/32,  3/32,  7/128,  1/32,...       = A000265/(A075101=2*A084623) =: Coll(n),

0,       1/4,   3/8,   3/8,  5/16, 15/64, 21/128,...              = (0 before A069834)/new,

-1/4,   -1/4,     0,   1/4, 25/64, 27/64,...

0,      -1/2,  -3/4, -9/16, -5/32,...

1/2,     1/2, -9/16, -13/8,...

0,      17/8, 51/16,...

-17/8, -17/8,...

0

The first column is A198631/(A006519?), essentially the fractional Euler numbers 1, -1/2, 0, 1/4, 0,...  in A060096.

Numerators b(n): 1, 1, 1, 0, 1, 1, -1, 1, 3, 1, ... .

Coll(n+1) - 2*Coll(n) = -1/2, -5/8, -1/2, -11/32, -7/32, -17/128, -5/64, -23/512, ... = -A075677/new, from Collatz problem.

There are three different Bernoulli numbers:

The first Bernoulli numbers are  1, -1/2, 1/6, 0,... = A027641(n)/A027642(n).

The second Bernoulli numbers are 1,  1/2, 1/6, 0,... = A164555(n)/A027642(n). These are the binomial transform of the first one.

The third Bernoulli numbers are  1,   0,  1/6, 0,... = A176327(n)/A027642(n), the half sum. Via A177427(n) and A191567(n), they yield the Balmer series A061037/A061038.

There are three different fractional Euler numbers:

1) The first are  1, -1/2, 0, 1/4, 0, -1/2,... in A060096(n).

Also Akiyama-Tanigawa algorithm for ( 1, 3/2, 7/4, 15/8, 31/16, 63/32,... = A000225(n+1)/A000079(n) ).

2) The second are 1, 1/2, 0, -1/4, 0,  1/2,... , mentioned by Wolfdieter Lang in  A198631(n).

3) The third are  0, 1/2, 0, -1/4, 0,  1/2,... , half difference of 2) and 1).

Also Akiyama-Tanigawa algorithm for ( 0, -1/2, -3/4, -7/8, -15/16, -31/32,... =  A000225(n)/A000079(n) ). See A097110(n).

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

a(n)=

1,

2, 2,

1, 2,  4,

4, 4,  8,  8,

1, 4,  8,  4, 16,

2, 2,  1,  8, 32, 32,

1, 2,  4,  4, 16, 32,  64,

8, 8, 16, 16, 64, 64, 128, 128,

MATHEMATICA

max = 10; t[0, k_] := 1/2^k; t[n_, k_] := t[n, k] = (k + 1)*(t[n - 1, k] - t[n - 1, k + 1]); denoms = Table[t[n, k] // Denominator, {n, 0, max}, {k, 0, max - n}]; Table[denoms[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Feb 05 2013 *)

CROSSREFS

Cf. Second Bernoulli numbers A164555(n)/A027642(n) via Akiyama-Tanigawa algorithm for 1/(n+1).

Sequence in context: A144218 A098691 A035364 * A143808 A247495 A230290

Adjacent sequences:  A209305 A209306 A209307 * A209309 A209310 A209311

KEYWORD

nonn,frac,tabl

AUTHOR

Paul Curtz, Jan 18 2013

STATUS

approved

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Last modified December 18 00:20 EST 2014. Contains 252040 sequences.