A130406 Column 1 of triangle A130405.

1, 3, 13, 83, 814, 12502, 303102, 11681388, 718217460, 70660085940, 11145552305760, 2823029266531680, 1149529177121700960, 753213189796615454400, 794745942920930023732800

Offset: 0

Author

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

Keywords

• nonn

Examples

			a(n) = A003266(n+1)*[F(n+1) + F(n+2)*[1+ 1/2+ 2/3+ 3/5+...+ F(n)/F(n+1)]]:
a(3) = 1*1*2*3*( 3 + 5*(1/1 + 1/2 + 2/3) ) = 83;
a(4) = 1*1*2*3*5*( 5 + 8*(1/1 + 1/2 + 2/3 + 3/5) ) = 814;
a(5) = 1*1*2*3*5*8*( 8 + 13*(1/1 + 1/2 + 2/3 + 3/5 + 5/8) ) = 12502.
		

Crossrefs

Cf. A130405, A130407, A003266, A000045.

Programs

PARI

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

PARI

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

PARI

a(n)=prod(i=1,n+1,fibonacci(i))*(fibonacci(n+1) + fibonacci(n+2)*sum(k=0,n,fibonacci(k)/fibonacci(k+1)))

Formula

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

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