A103452 Inverse of number triangle A103451.

1, -1, 1, -1, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Paul Barry, Feb 06 2005

Keywords

• easy
• sign
• tabl

Comments

Triangle T(n,k), 0 <= k <= n, read by rows given by [ -1, 2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 01 2007

Examples

			Triangle begins
   1;
  -1, 1;
  -1, 0, 1;
  -1, 0, 0, 1;
  -1, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
Production matrix begins
  -1, 1;
  -2, 1, 1;
  -2, 1, 0, 1;
  -2, 1, 0, 0, 1;
  -2, 1, 0, 0, 0, 1;
  -2, 1, 0, 0, 0, 0, 1;
  -2, 1, 0, 0, 0, 0, 0, 1;
  -2, 1, 0, 0, 0, 0, 0, 0, 1;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10010 (Rows 0 <= n <= 140)

Crossrefs

Cf. A000007 (row sums), A103453, A103454, A103455.

Programs

Mathematica

Table[Range[n] /. {k_ /; k == 1 && n != 1 -> -1, k_ /; k == n -> 1, Integer -> 0}, {n, 15}] // Flatten (* _Michael De Vlieger, Jul 21 2016 *)

Sage

flatten([[1 if k==n else -1 if k==0 else 0 for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 18 2021

Formula

T(n,k) = 1 if k = n, -1 if k = 0, otherwise 0.

Sum_{k=0..n} T(n, k) = 0^n (row sums).

Sum_{k=0..floor(n/2)} T(n-k, k) = 0^n - (1-(-1)^n)/2 (diagonal sums).

G.f.: (1 - 2*x + y*x^2)/((1-x)*(1-y*x)). - Philippe Deléham, Feb 11 2012