0,17

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.

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

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).

Sequence in context: A127324 A083917 A117974 this_sequence A156064 A158757 A085983

Adjacent sequences: A156059 A156060 A156061 this_sequence A156063 A156064 A156065

easy,nonn,tabl

Paul Barry (pbarry(AT)wit.ie), Oct 20 2009