A111963 Inverse of renewal array for central trinomial numbers.

1, -1, 1, -1, -2, 1, 3, -1, -3, 1, 1, 8, 0, -4, 1, -9, -3, 14, 2, -5, 1, 1, -26, -15, 20, 5, -6, 1, 27, 27, -45, -37, 25, 9, -7, 1, -13, 76, 98, -56, -70, 28, 14, -8, 1, -81, -135, 108, 228, -46, -114, 28, 20, -9, 1, 67, -202, -459, 48, 420, 0, -168, 24, 27, -10, 1, 243, 567, -135, -1035, -210, 662, 98, -230, 15, 35, -11, 1, -285

Offset: 0

Author

Paul Barry, Aug 23 2005

Keywords

• easy
• sign
• tabl

Comments

Row sums have g.f. 1/sqrt(1+4x^2) [alternating sign central binomial numbers with interpolated zeros]. Diagonal sums are A111964. Inverse of A111960. Factors as (1/sqrt(1+4x^2),x/sqrt(1+4x^2))*(1/(1+x),x/(1+x)).

Examples

			Triangle begins
1;
-1,1;
-1,-2,1;
3,-1,-3,1;
1,8,0,-4,1;
-9,-3,14,2,-5,1;
1,-26,-15,20,5,-6,1;
		

Formula

Riordan array (1/(sqrt(1+4x^2)+x), x/(sqrt(1+4x^2)+x)); Number triangle T(n, k)=sum{i=0..floor(n/2), C(2i+k-n-1, k)*C((2i-n-1)/2, i)(-1)^n*4^i}.