OFFSET
0,2
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alfred Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 17.
Nadia Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Thomas Koshy, Infinite Sums Involving Jacobsthal Polynomial Products Revisited, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), pp. 3-14.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).
FORMULA
a(n) = ((4+n, 4)) (see A010048), or fibonomial(4+n, 4).
G.f.: 1/(1-5*x-15*x^2+15*x^3+5*x^4-x^5) = 1/((1-x)*(1+3*x+x^2)*(1-7*x+x^2)) (see Comments to A055870). a(n)= 7*a(n-1)-a(n-2)+((-1)^n)*fibonomial(n+2, 2), n >= 2; a(0)=1, a(1)=5; fibonomial(n+2, 2)= A001654(n+1).
a(n) = Product_{k=1..n} Fibonacci(k+4)/Fibonacci(k). - Gary Detlefs, Feb 06 2011
a(n) = (F(n+3)^2-F(n+2)^2)*F(n+3)*F(n+2)/6, where F(n) is the n-th Fibonacci number. - Gary Detlefs, Oct 12 2011
a(n) = a(-5-n) for all n in Z. - Michael Somos, Sep 19 2014
0 = a(n)*(+a(n+1) - 2*a(n+2)) + a(n+1)*(-5*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 19 2014
From Peter Bala, Mar 30 2015: (Start)
The o.g.f. A(x) = 1/(1 - 5*x - 15*x^2 + 15*x^3 + 5*x^4 - x^5). Hence A(x) (mod 25) = 1/(1 - 5*x + 10*x^2 - 10^x^3 + 5*x^4 - x^5) (mod 25) = 1/(1 - x)^5 (mod 25). It follows by Theorem 1 of Heninger et al. that A(x)^(1/5) = 1 + x + 6*x^2 + 26*x^3 + ... has integral coefficients.
Sum_{n >= 0} a(n)*x^n = exp( Sum_{n >= 1} Fibonacci(5*n)/Fibonacci(n)*x^n/n ). Cf. A084175, A099930. (End)
Sum_{n>=0} 1/a(n) = 51/2 - 15*phi, where phi is the golden ratio (A001622) (Koshy, 2022, section 3.3, p. 9). - Amiram Eldar, Jan 23 2025
EXAMPLE
G.f. = 1 + 5*x + 40*x^2 + 260*x^3 + 1820*x^4 + 12376*x^5 + 85085*x^6 + ... .
MAPLE
with (combinat): a:=n->1/6*fibonacci(n)*fibonacci(n+1)*fibonacci(n+2)*fibonacci(n+3): seq(a(n), n=1..18); # Zerinvary Lajos, Oct 07 2007
A001656:=-1/(z-1)/(z**2-7*z+1)/(z**2+3*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(Fibonacci[n+3]*Fibonacci[n+2]*Fibonacci[n+1]*Fibonacci[n])/6, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
LinearRecurrence[{5, 15, -15, -5, 1}, {1, 5, 40, 260, 1820}, 20] (* Vincenzo Librandi, Aug 02 2012 *)
Times@@@Partition[Fibonacci[Range[30]], 4, 1]/6 (* Harvey P. Dale, Oct 13 2016 *)
PROG
(PARI) b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 4)) \\ Joerg Arndt, May 08 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Corrected and extended by Wolfdieter Lang, Jun 27 2000
More terms from Vladimir Joseph Stephan Orlovsky, Nov 23 2009
STATUS
approved