A156062 Riordan array (1/(1-x^4), x/(1-x^4)).

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 1, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 3, 0, 0, 0, 6, 0, 0, 0, 1, 0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 10, 0, 0, 0, 8, 0, 0, 0, 1, 1, 0, 0, 0, 15, 0, 0, 0, 9, 0, 0, 0, 1, 0, 4, 0, 0, 0, 21, 0, 0, 0, 10, 0, 0, 0

Offset: 0

Author

Paul Barry, Oct 20 2009

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A003269(n+1). Diagonal sums are the aerated Fibonacci numbers; thus

F(n+1)=sum{k=0..n, C((n+k)/2,k)*(1+(-1)^(n-k))/2}. Inverse is A156064.

Examples

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 0, 0, 1,
1, 0, 0, 0, 1,
0, 2, 0, 0, 0, 1,
0, 0, 3, 0, 0, 0, 1,
0, 0, 0, 4, 0, 0, 0, 1,
1, 0, 0, 0, 5, 0, 0, 0, 1,
0, 3, 0, 0, 0, 6, 0, 0, 0, 1,
0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 1,
0, 0, 0, 10, 0, 0, 0, 8, 0, 0, 0, 1,
1, 0, 0, 0, 15, 0, 0, 0, 9, 0, 0, 0, 1
Production matrix of this array is
0, 1,
0, 0, 1,
0, 0, 0, 1,
1, 0, 0, 0, 1,
0, 1, 0, 0, 0, 1,
0, 0, 1, 0, 0, 0, 1,
0, 0, 0, 1, 0, 0, 0, 1,
-3, 0, 0, 0, 1, 0, 0, 0, 1,
0, -3, 0, 0, 0, 1, 0, 0, 0, 1,
0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,
0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,
15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,
0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,
0, 0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,
0, 0, 0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,
-91, 0, 0, 0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1
where 1,1,-3,15,-91,612,.... is (-1)^(n-1)*C(4n-1,n)/(4n-1) (see A006632).
		

Formula

Triangle T(n,k)=C((n+3k)/4,k)((1+(-1)^(n-k))/2+cos(pi*(n-k)/2))/2.