A209308 Denominators of the Akiyama-Tanigawa algorithm applied to 2^(-n), written by antidiagonals.

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

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, the Oresme numbers n/2^n. Paul Curtz, Jan 18 2013 and May 11 2016

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

G. C. Greubel, Table of n, a(n) for n = 0..5049

A. F. Horadam, Oresme Numbers, Fibonacci Quarterly, 12, #3, 1974, pp. 267-271.

Example

Triangle begins:
  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-François Alcover, Feb 05 2013 *)

Crossrefs

Cf. Second Bernoulli numbers A164555(n)/A027642(n) via Akiyama-Tanigawa algorithm for 1/(n+1), A272263.

Sequence in context: A324251 A035364 A261734 * A143808 A294600 A247495

Adjacent sequences: A209305 A209306 A209307 * A209309 A209310 A209311

Keyword

nonn,frac,tabl

Author

Paul Curtz, Jan 18 2013

Status

approved