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

Author

Paul Curtz, May 21 2014

Keywords

• sign
• frac
• tabl

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.

Examples

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

Programs

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 *)