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

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

0,7

Comments

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.

Links

Table of n, a(n) for n=0..54.

Example

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

Mathematica

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 *)

Crossrefs

Sequence in context: A025254 A245537 A277198 * A177058 A176921 A000503

Adjacent sequences: A242732 A242733 A242734 * A242736 A242737 A242738

Keyword

sign,frac,tabl

Author

Paul Curtz, May 21 2014

Status

approved