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URL: https://oeis.org/A130407

⇱ A130407 - OEIS

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A130407
A diagonal of triangle A130405.
2
1, 3, 9, 37, 233, 2254, 34342, 827262, 31730508, 1943441460, 190609515540, 29988517246560, 7579307667005280, 3080578207713982560, 2015291663362285214400, 2123462159890867147060800
OFFSET
0,2
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.
EXAMPLE
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.
PROG
(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)) )
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 24 2007, May 25 2007
STATUS
approved