A130407 A diagonal of triangle A130405.

1, 3, 9, 37, 233, 2254, 34342, 827262, 31730508, 1943441460, 190609515540, 29988517246560, 7579307667005280, 3080578207713982560, 2015291663362285214400, 2123462159890867147060800

Offset: 0

Author

Paul D. Hanna, May 24 2007, May 25 2007

Keywords

• nonn

Examples

			a(n) = A003266(n)*[F(n+2) + F(n+1)*[1+ 2/1+ 3/2+ 5/3+...+ F(n+1)/F(n)]]:
a(3) = 1*1*2*( 5 + 3*(1/1 + 2/1 + 3/2) ) = 37;
a(4) = 1*1*2*3*( 8 + 5*(1/1 + 2/1 + 3/2 + 5/3) ) = 233;
a(5) = 1*1*2*3*5*( 13 + 8*(1/1 + 2/1 + 3/2 + 5/3 + 8/5) ) = 2254.
		

Crossrefs

Cf. A130405, A130406, A003266, A000045.

Programs

PARI

a(n)=polcoeff(prod(i=0,n+1,fibonacci(i+1)+x*fibonacci(i)),n)

PARI

/* Recurrence a(n) = F(n+1)*a(n-1) + F(n+2)*A003266(n): */ {a(n)=if(n==0,1,fibonacci(n+1)*a(n-1)+fibonacci(n+2)*prod(i=1,n,fibonacci(i)))}

PARI

a(n)=prod(i=1,n,fibonacci(i))*(fibonacci(n+2) + fibonacci(n+1)*sum(k=1,n,fibonacci(k+1)/fibonacci(k)) )

Formula

a(n) = F(n+1)*a(n-1) + F(n+2)*A003266(n), where A003266(n) is the product of the first n nonzero Fibonacci numbers (A000045) and F(n) = A000045(n).

a(n) = A003266(n)*[ F(n+2) + F(n+1)*Sum_{k=0..n} F(k+1)/F(k) ] where F(n)=A000045(n) is the n-th Fibonacci number.