login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001658 Fibonomial coefficients.
(Formerly M4919 N2112)
6
1, 13, 273, 4641, 85085, 1514513, 27261234, 488605194, 8771626578, 157373300370, 2824135408458, 50675778059634, 909348684070099, 16317540120588343, 292806787575013635, 5254201798026392211, 94282845030238533383, 1691836875411111866723, 30358781826262552258596 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

It appears a(n) = 13a(n-1) + 104a(n-2) - 260a(n-3) - 260a(n-4) + 104a(n-5) + 13a(n-6) - a(n-7) for n > 6. - John W. Layman, Apr 14 2000

Layman's formula is correct. - Wolfdieter Lang, Jul 13 2000

Layman's formula is a consequence of formula 2.8 (p. 116) of Lind (1971). - Dale Gerdemann, May 08 2016

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.

A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 74.

D. A. Lind, A Determinant Involving Binomial Coefficients, Part 1, Part 2, Fibonacci Quarterly 9.2, 1971.

Index entries for linear recurrences with constant coefficients, signature (13, 104, -260, -260, 104, 13, -1).

FORMULA

From Wolfdieter Lang, Jul 13 2000: (Start)

G.f.: 1/(1-13*x-104*x^2+260*x^3+260*x^4-104*x^5-13*x^6+x^7) = 1/((1+x)*(1-3*x+x^2)*(1+7*x+x^2)*(1-18*x+x^2)) (see Comments to A055870).

a(n) = 5*a(n-1)+F(n-5)*Fibonomial(n+5, 5), n >= 1, a(0) = 1; F(n) = A000045(n) (Fibonacci). a(n) = 18*a(n-1)-a(n-2)+((-1)^n)*Fibonomial(n+4, 4), n >= 2; a(0) = 1, a(1) = 13; Fibonomial(n+4, 4) = A001656(n). (End)

From Gary Detlefs, Dec 03 2012: (Start)

a(n) = F(n+1)*F(n+2)*F(n+3)*F(n+4)*F(n+5)*F(n+6)/240.

a(n) = (F(n+5)^2 - F(n+4)^2)*(F(n+3)^4 - 1)/240, where F(n) = A000045(n). (End)

Conjecture: a(n) = F(7)^(n-6) + Sum_{i=3..n-5} F(i-2)F(6)^{i-1}F(7)^{n-i-5} + Sum_{j=3..i} F(i-2)F(j-2)F(5)^{j-1}F(6)^{i-j}F(7)^{n-i-5} + Sum_{k=3..j} F(i-2)F(j-2)F(k-2)F(4)^{k-1}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5} + Sum_{l=3..k} F(i-2)F(j-2)F(k-2)F(l-2)F(3)^{l-1}F(4)^{k-l}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5} + Sum_{m=3..l} F(i-2)F(j-2)F(k-2)F(l-2)F(m-2)F(m)F(3)^{l-m}F(4)^{k-l}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5}, where F(n)=A000045(n). - Dale Gerdemann, May 08 2016

MAPLE

with(combinat):a:=n->1/240*fibonacci(n)*fibonacci(n+1)*fibonacci(n+2)*fibonacci(n+3)*fibonacci(n+4)*fibonacci(n+5): seq(a(n), n=1..17); # Zerinvary Lajos, Oct 07 2007

MATHEMATICA

f[n_] := Times @@ Fibonacci[Range[n+1, n+6]]/240; Table[f[n], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)

LinearRecurrence[{13, 104, -260, -260, 104, 13, -1}, {1, 13, 273, 4641, 85085, 1514513, 27261234}, 20] (* Harvey P. Dale, Aug 24 2014 *)

PROG

(PARI) b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));

vector(20, n, b(n-1, 6))  \\ Joerg Arndt, May 08 2016

CROSSREFS

Cf. A001654, A001656, A001657.

Sequence in context: A142262 A163155 A183515 * A034911 A196652 A197455

Adjacent sequences:  A001655 A001656 A001657 * A001659 A001660 A001661

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Wolfdieter Lang, Jul 13 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified October 18 10:28 EDT 2017. Contains 293507 sequences.