A110324 Inverse of a number triangle related to the Jacobsthal numbers.

1, -1, 1, -4, -2, 1, 0, -12, -3, 1, 0, 0, -24, -4, 1, 0, 0, 0, -40, -5, 1, 0, 0, 0, 0, -60, -6, 1, 0, 0, 0, 0, 0, -84, -7, 1, 0, 0, 0, 0, 0, 0, -112, -8, 1, 0, 0, 0, 0, 0, 0, 0, -144, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, -180, -10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -220, -11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -264, -12, 1

Offset: 0

Author

Paul Barry, Jul 20 2005

Keywords

• easy
• sign
• tabl

Comments

Row sums are A110325. Diagonal sums are A110326. Inverse of A110321. The result can be generalized as follows: The triangle whose columns have e.g.f. (x^k/k!)/(1-a*x-b*x^2) has inverse T(n,k)=if(n=k,1,if(n-k=1,-a*binomial(n,1),if(n-k=2,-2*b*binomial(n,2),0)))

Examples

			Rows begin
1;
-1,1;
-4,-2,1;
0,-12,-3,1;
0,0,-24,-4,1;
0,0,0,-40,-5,1;
0,0,0,0,-60,-6,1;
		

Formula

T(n, k)=if(n=k, 1, if(n-k=1, -binomial(n, 1), if(n-k=2, -4*binomial(n, 2), 0)))