A128064 Triangle T with T(n,n)=n, T(n,n-1)=-(n-1) and otherwise T(n,k)=0; 0

1, -1, 2, 0, -2, 3, 0, 0, -3, 4, 0, 0, 0, -4, 5, 0, 0, 0, 0, -5, 6, 0, 0, 0, 0, 0, -6, 7, 0, 0, 0, 0, 0, 0, -7, 8, 0, 0, 0, 0, 0, 0, 0, -8, 9, 0, 0, 0, 0, 0, 0, 0, 0, -9, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 12

Offset: 1

Author

Gary W. Adamson, Feb 14 2007

Keywords

• tabl
• sign
• easy

Comments

The matrix inverse = (1/1; 1/2, 1/2; 1/3, 1/3, 1/3;...). Binomial transform of A128064 = A128065. A128064 * A007318 = A103406.

The positive version with row sums 2n+1 is given by T(n,k)=sum{j=k..n, C(n,j)*C(j,k)*(-1)^(n-j)*(j+1)}. - Paul Barry, May 26 2007

Binomial transform of unsigned sequence is A003506. - Gary W. Adamson, Aug 29 2007

Table T(n,k) read by antidiagonals. T(n,1) = n (for n>1), T(n,2) = -n, T(n,k) = 0, k > 2. - Boris Putievskiy, Feb 07 2013

Examples

			First few rows of the triangle are:
1;
-1,2;
0,-2,3;
0,0,-3,4;
0,0,0,-4,5;
0,0,0,0,-5,6;
0,0,0,0,0,-6,7;
...
From _Boris Putievskiy_, Feb 07 2013: (Start)
The start of the sequence as table:
1..-1..0..0..0..0..0...
2..-2..0..0..0..0..0...
3..-3..0..0..0..0..0...
4..-4..0..0..0..0..0...
5..-5..0..0..0..0..0...
6..-6..0..0..0..0..0...
7..-7..0..0..0..0..0...
. . .
(End)
		

Links

• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A128065, A103406, A003506, A002260, A167374.

Programs

Mathematica

row[1] = {1}; row[2] = {-1, 2}; row[n_] := Join[Array[0&, n-2], {-n+1, n}]; Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 12 2015 *)

Formula

Number triangle T(n,k)=sum{j=k..n, C(n,j)*C(j,k)*(-1)^(j-k)*(j+1)}. - Paul Barry, May 26 2007

a(n) = A002260(n)*A167374(n); a(n) = i*floor((i+2)/(t+2))*(-1)^(i+t+1), where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 07 2013

G.f.: (-1)^k*[x^k*exp(k*x)]'/exp(k*x)=sum(n>=k, (-1)^n*T(n,k)*x^n). - Vladimir Kruchinin, Oct 18 2013