A001657 Fibonomial coefficients: column 5 of A010048. (Formerly M4568 N1945)

1, 8, 104, 1092, 12376, 136136, 1514513, 16776144, 186135312, 2063912136, 22890661872, 253854868176, 2815321003313, 31222272414424, 346260798314872, 3840089017377228, 42587248616222024, 472299787252290712, 5237885063192296801, 58089034826620525728

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

T. D. Noe, Table of n, a(n) for n = 0..200

A. 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.

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 (8,40,-60,-40,8,1).

Formula

a(n) = A010048(5+n, 5) (or fibonomial(5+n, 5)).

G.f.: 1/(1-8*x-40*x^2+60*x^3+40*x^4-8*x^5-x^6) = 1/((1-x-x^2)*(1+4*x-x^2)*(1-11*x-x^2)) (see Comments to A055870).

a(n) = 11*a(n-1) + a(n-2) + ((-1)^n)*fibonomial(n+3, 3), n >= 2; a(0)=1, a(1)=8; fibonomial(n+3, 3)= A001655(n).

a(n) = Fibonacci(n+3)*(Fibonacci(n+3)^4-1)/30. - Gary Detlefs, Apr 24 2012

a(n) = (A049666(n+3) + 2*(-1)^n*A001076(n+3) - 3*A000045(n+3))/150, n >= 0, with A049666(n) = F(5*n)/5, A001076(n) = F(3*n)/2 and A000045(n) = F(n). From the partial fraction decomposition of the o.g.f. and recurrences. - Wolfdieter Lang, Aug 23 2012

a(n) = a(-6-n) * (-1)^n for all n in Z. - Michael Somos, Sep 19 2014

0 = a(n)*(-a(n+1) - 3*a(n+2)) + a(n+1)*(-8*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 19 2014

Example

G.f. = 1 + 8*x + 104*x^2 + 1092*x^3 + 12376*x^4 + 136136*x^5 + 1514513*x^6 + ...

Maple

with(combinat) : a:=n-> 1/30*fibonacci(n)*fibonacci(n+1)*fibonacci(n+2)*fibonacci(n+3)*fibonacci(n+4): seq(a(n), n=1..19); # Zerinvary Lajos, Oct 07 2007
A001657:=-1/(z**2+11*z-1)/(z**2-4*z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation

Mathematica

f[n_] := Times @@ Fibonacci[Range[n + 1, n + 5]]/30; t = Table[f[n], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
LinearRecurrence[{8, 40, -60, -40, 8, 1}, {1, 8, 104, 1092, 12376, 136136}, 20] (* Harvey P. Dale, Nov 30 2019 *)

Prog

(PARI) a(n)=(n->(n^5-n)/30)(fibonacci(n+3)) \\ Charles R Greathouse IV, Apr 24 2012
(PARI) b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 5))  \\ Joerg Arndt, May 08 2016

Crossrefs

Cf. A010048, A001654-A001658, A065563.

Sequence in context: A164760 A335608 A109774 * A282185 A354064 A106260

Adjacent sequences: A001654 A001655 A001656 * A001658 A001659 A001660

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Corrected and extended by Wolfdieter Lang, Jun 27 2000

Status

approved