A206771 -id:A206771 - cp's OEIS frontend

A231234 Denominators related to A206771 and Lorentz gamma factor.

1, 1, 1, 8, 4, 128, 128, 1024, 256, 32768, 32768, 262144, 131072, 4194304, 4194304, 33554432, 4194304, 2147483648, 2147483648, 17179869184, 8589934592, 274877906944, 274877906944, 2199023255552, 549755813888, 70368744177664

Offset: 0

Jean-François Alcover and Paul Curtz, Nov 06 2013

See A206771.

In addition, it can be noticed that a(n) is always a power of 2 and that a(2n-1)/a(2n) is A006519(n).

Cf. A001803, A001790, A006519, A046161, A206771.

max = 25; A001803 = CoefficientList[Series[(1 - x)^(-3/2), {x, 0, max}], x] // Numerator; A001790 = CoefficientList[Series[1/Sqrt[(1 - x)], {x, 0, max}], x] // Numerator; A046161 = Table[Binomial[2 n, n]/4^n, {n, 0, max}] // Denominator; a[0] = 1;  a[n_] := (A001803[[n]] + A001790[[n]])/(2*A046161[[n]]) // Denominator; Table[a[n], {n, 0, max}]
(* or, directly: *) a[0] = 1; a[n_] := Denominator[4^(1-n)*Binomial[2*n-2, n-1]]/2^IntegerExponent[n, 2]; Table[a[n], {n, 0, max}]

a(n) = denominator(4^(1-n)*binomial(2*n-2, n-1))/2^valuation(n, 2) (where valuation(n,2) = A007814(n)).
a(n) = 2^(2*n-2-adic valuation(n, 2)-valuation(binomial(2*n-2, n-1), 2)).
a(n) = A046161(n-1)/A006519(n).

A224270 Absolute values of the numerators of the third column of ( 0 followed by (interleave 0 , A001803(n))/A060818(n) ) and its successive differences.

Original entry on oeis.org

1, 1, 5, 11, 95, 203, 861, 1815, 30459, 63635, 264979, 550069, 4555915, 9412543, 38816525, 79898895, 2627302995, 5392044675, 22104436695, 45256266825, 370241638305, 756514878405, 3088866211275, 6300861570705, 102746354288175, 209286947903319

Offset: 0

Paul Curtz, Apr 02 2013

The array is
  0,          0,      1,     0,   3/2,     0,  15/8, 0,...
  0,          1,     -1,   3/2,  -3/2,  15/8, -15/8,...
  1,         -2,    5/2,    -3,  27/8, -15/4,...
  -3,       9/2,  -11/2,  51/8, -57/8,...
  15/2,     -10,   95/8, -27/2,...
  -35/2,  175/8, -203/8,...
  315/8, -189/4,...
  -693/8,...
Note A001803 in the first column and a variant of A206771(n) in the second column.
Now consider a(n)/A046161(n) and its differences:
  1,            1/2,     5/8,  11/16,  95/128, 203/256, 861/1024,...
  -1/2,         1/8,    1/16,  7/128,  13/256, 49/1024,...   =b(n)/A046161(n)
  5/8,        -1/16,  -1/128, -1/256, -3/1024,...
  -11/16,     7/128,   1/256, 1/1024,...
  95/128,   -13/256, -3/1024,...
  -203/256, 49/1024,...
  861/1024,...
This an autosequence of second kind. The first column is the signed sequence.
(Its companion, the corresponding autosequence of first kind, is 0, 1, 1, 9/8, 5/4,... in A206771).
Main diagonal: 1, 1/8, -1/128,... = A002596(n)/A061549(n) ?
b(n) = a(n+1) - A171977*a(n). Also for two successive rows (with shifted A171977).

			a(n)=numerators of 0+1=1, 0+1/2=1/2, 1/4+3/8=5/8, 3/8+5/16=11/16, 15/32+35/128=95/128,... .

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001790, A001803, A002596, A046161, A061549, A098597, A171977, A206771.

nmax = 25; t1 = Table[ Numerator[ (2*n+1)*(Binomial[2*n, n]/4^n)] / Denominator[ Binomial[2*n, n]/4^n], {n, 0, Ceiling[nmax/2]}]; t2 = Join[{0}, Table[ If[ OddQ[n], 0, t1[[n/2]] ], {n, 1, nmax+2}] ]; t3 = Table[ Differences[t2, n], {n, 0, nmax}]; t3[[All, 3]] // Numerator // Abs (* Jean-François Alcover, Apr 02 2013 *)

Numerators of (0, 0 followed by A001803(n)/(4*A046161(n))) + A001790(n)/A046161(n).

More terms from Jean-François Alcover, Apr 02 2013

A242735 Array read by antidiagonals: form difference table of the sequence of rationals 0, 0 followed by A001803(n)/A046161(n), then extract numerators.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, -3, -1, 1, 3, 15, 3, -1, 3, 15, -35, -5, 1, -1, 5, 35, 315, 35, -5, 3, -5, 35, 315, -693, -63, 7, -3, 3, -7, 63, 693, 3003, 231, -21, 7, -5, 7, -21, 231, 3003, -6435, -429, 33, -9, 5, -5, 9, -33, 429, 6435

Offset: 0

Paul Curtz, May 21 2014

Difference table of c(n)/d(n) = 0, 0, followed by A001803(n)/A046161(n):
0,           0,      1,    3/2,    15/8,    35/16,   315/128, ...
0,           1,    1/2,    3/8,    5/16,   35/128,    63/256, ...
1,        -1/2,   -1/8,  -1/16,  -5/128,   -7/256,  -21/1024, ...
-3/2,      3/8,   1/16,  3/128,   3/256,   7/1024,    9/2048, ...
15/8,    -5/16, -5/128, -3/256, -5/1024,  -5/2048, -45/32768, ...
-35/16, 35/128,  7/256, 7/1024,  5/2048, 35/32768,  35/65536, ... etc.
d(n) = 1, 1, followed by A046161(n).
c(n)/d(n) is an autosequence (a sequence whose inverse binomial transform is the signed sequence) of the second kind (the main diagonal is equal to the first upper diagonal multiplied by 2). See A187791.
Antidiagonal denominators: repeat n+1 times d(n).
Second row without 0: Lorentz (gamma) factor = A001790(n)/A046161(n).
Third row: Lorentz beta factor = 1 followed by -A098597(n). Lorbeta(n) in A206771.

			a(n) as a triangle:
   0;
   0,  0;
   1,  1,  1;
  -3, -1,  1,  3;
  15,  3, -1,  3, 15;
  etc.

c[n_] := (2*n-3)*Binomial[2*(n-2), n-2]/4^(n-2) // Numerator; d[n_] := Binomial[2*(n-2), n-2]/4^(n-2) // Denominator; Clear[a]; a[0, k_] := c[k]/d[k]; a[n_, k_] := a[n, k] = a[n-1, k+1] - a[n-1, k]; Table[a[n-k, k] // Numerator, {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 17 2014 *)

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A231234 Denominators related to A206771 and Lorentz gamma factor.

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A224270 Absolute values of the numerators of the third column of ( 0 followed by (interleave 0 , A001803(n))/A060818(n) ) and its successive differences.

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A242735 Array read by antidiagonals: form difference table of the sequence of rationals 0, 0 followed by A001803(n)/A046161(n), then extract numerators.

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