A119337 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i-k, C(k,3j)*C(i-k,3j)}}.

1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, -1, 3, -3, 1, 0, 1, -4, 6, -4, 1, 0, -1, 5, -9, 10, -5, 1, 0, 1, -6, 12, -16, 15, -6, 1, 0, -1, 7, -15, 19, -25, 21, -7, 1, 0, 1, -8, 18, -16, 20, -36, 28, -8, 1, 0, -1, 9, -21, 4, 24, 6, -49, 36, -9, 1

Offset: 0

Author

Paul Barry, May 14 2006

Keywords

• easy
• sign
• tabl

Comments

Row sums have g.f. (1+x)/(1-x)^6. Multiply by Pascal's triangle A007318 to get A119335.

Examples

			Triangle begins
1,
0, 1,
0, -1, 1,
0, 1, -2, 1,
0, -1, 3, -3, 1,
0, 1, -4, 6, -4, 1,
0, -1, 5, -9, 10, -5, 1,
0, 1, -6, 12, -16, 15, -6, 1,
0, -1, 7, -15, 19, -25, 21, -7, 1,
0, 1, -8, 18, -16, 20, -36, 28, -8, 1,
0, -1, 9, -21, 4, 24, 6, -49, 36, -9, 1
		

Formula

Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,3j)x^(3j)}