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

%I #2 Mar 30 2012 18:59:22

%S 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,

%T 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,

%U 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

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

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

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

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

%e Triangle begins

%e 1,

%e 0, 1,

%e 0, 0, 1,

%e 0, 0, 0, 1,

%e 1, 0, 0, 0, 1,

%e 0, 2, 0, 0, 0, 1,

%e 0, 0, 3, 0, 0, 0, 1,

%e 0, 0, 0, 4, 0, 0, 0, 1,

%e 1, 0, 0, 0, 5, 0, 0, 0, 1,

%e 0, 3, 0, 0, 0, 6, 0, 0, 0, 1,

%e 0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 1,

%e 0, 0, 0, 10, 0, 0, 0, 8, 0, 0, 0, 1,

%e 1, 0, 0, 0, 15, 0, 0, 0, 9, 0, 0, 0, 1

%e Production matrix of this array is

%e 0, 1,

%e 0, 0, 1,

%e 0, 0, 0, 1,

%e 1, 0, 0, 0, 1,

%e 0, 1, 0, 0, 0, 1,

%e 0, 0, 1, 0, 0, 0, 1,

%e 0, 0, 0, 1, 0, 0, 0, 1,

%e -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e 0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e 0, 0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e 0, 0, 0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1,

%e -91, 0, 0, 0, 15, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, 1

%e where 1,1,-3,15,-91,612,.... is (-1)^(n-1)*C(4n-1,n)/(4n-1) (see A006632).

%K easy,nonn,tabl

%O 0,17

%A _Paul Barry_, Oct 20 2009