A062073 -id:A062073 - cp's OEIS frontend

A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

1, 1, 1, 2, 6, 30, 240, 3120, 65520, 2227680, 122522400, 10904493600, 1570247078400, 365867569267200, 137932073613734400, 84138564904377984000, 83044763560621070208000, 132622487406311849122176000, 342696507457909818131702784000

Offset: 0

N. J. A. Sloane

Equals right border of unsigned triangle A158472. - Gary W. Adamson, Mar 20 2009
Three closely related sequences are A194157 (product of first n nonzero F(2*n)), A194158 (product of first n nonzero F(2*n-1)) and A123029 (a(2*n) = A194157(n) and a(2*n-1) = A194158(n)). - Johannes W. Meijer, Aug 21 2011
a(n+1)^2 is the number of ways to tile this pyramid of height n with squares and dominoes, where vertical dominoes can only appear (if at all) in the central column. Here is a pyramid of height n=4,
       _
     ||_
   ||_||
 ||_|||_|_
|||_|||_|_|,
and here is one of the a(5)^2 = 900 possible such tilings with our given restrictions:
       _
     | |
   ||_||
 |__|_|_|_
||__|___|||. - Greg Dresden and Jiayi Liu, Aug 23 2024

			a(5) = 30 because the first 5 Fibonacci numbers are 1, 1, 2, 3, 5 and 1 * 1 * 2 * 3 * 5 = 30.
a(6) = 240 because 8 is the sixth Fibonacci number and a(5) * 8 = 240.
a(7) = 3120 because 13 is the seventh Fibonacci number and a(6) * 13 = 3120.
G.f. = 1 + x + x^2 + 2*x^3 + 6*x^4 + 30*x^5 + 240*x^6 + 3120*x^7 + ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second edition, Addison Wesley, p 597
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..99 (terms n = 1..50 from T. D. Noe)
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 74.
Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.
Shyam Sunder Gupta, Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
Tipaluck Krityakierne and Thotsaporn Aek Thanatipanonda, Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C^2-finite Sequences, in Applied Mathematical Analysis and Computations (SGMC 2021) Springer Proc. Math. Stat., Vol. 471. Springer, Cham, 255-297. See p. 287.
Mathematica Stack Exchange, Product of Fibonacci numbers using For/Do/While loops.
Yuri V. Matiyasevich and Richard K. Guy, A new formula for Pi, Amer. Math. Monthly 93 (1986), no. 8, 631-635. Math. Rev. 2000i:11199.
Aidan Sudbury, Arthur Sun, David Treeby, and Edward Wang, Pick-up Sticks and the Fibonacci Factorial, arXiv:2504.19911 [math.PR], 2025.
Thotsaporn Aek Thanatipanonda and Yi Zhang, Sequences: Polynomial, C-finite, Holonomic, ..., arXiv:2004.01370 [math.CO], 2020. See pp. 5-6.
Eric Weisstein's World of Mathematics, Fibonorial
Index to divisibility sequences.

Cf. A000045, A101689, A135598, A158472.
Cf. A123741 (for Fibonacci second version), A002110 (for primes), A070825 (for Lucas), A003046 (for Catalan), A126772 (for Padovan), A069777 (q-factorial numbers for sums of powers). - Johannes W. Meijer, Aug 21 2011
Cf. A176343, A238243, A238244.

a003266 n = a003266_list !! (n-1)
a003266_list = scanl1 (*) $ tail a000045_list
-- Reinhard Zumkeller, Sep 03 2013

with(combinat): A003266 := n-> mul(fibonacci(i),i=1..n): seq(A003266(n), n=0..20);

Rest[FoldList[Times,1,Fibonacci[Range[20]]]] (* Harvey P. Dale, Jul 11 2011 *)
a[ n_] := If[ n < 0, 0, Fibonorial[n]]; (* Michael Somos, Oct 23 2017 *)
Table[Round[GoldenRatio^(n(n-1)/2) QFactorial[n, GoldenRatio-2]], {n, 20}] (* Vladimir Reshetnikov, Sep 14 2016 *)

a(n)=prod(i=1,n,fibonacci(i)) \\ Charles R Greathouse IV, Jan 13 2012

from itertools import islice
def A003266_gen(): # generator of terms
    a,b,c = 1,1,1
    while True:
        yield c
        c *= a
        a, b = b, a+b
A003266_list = list(islice(A003266_gen(),20)) # Chai Wah Wu, Jan 11 2023

a(n) is asymptotic to C*phi^(n*(n+1)/2)/sqrt(5)^n where phi = (1 + sqrt(5))/2 is the golden ratio and the decimal expansion of C is given in A062073. - Benoit Cloitre, Jan 11 2003
a(n+3) = a(n+1)*a(n+2)/a(n) + a(n+2)^2/a(n+1). - Robert Israel, May 19 2014
a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 06 2014
0 = a(n)*(+a(n+1)*a(n+3) - a(n+2)^2) + a(n+2)*(-a(n+1)^2) for all n >= 0. - Michael Somos, Oct 06 2014
Sum_{n>=1} 1/a(n) = A101689. - Amiram Eldar, Oct 27 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = A135598. - Amiram Eldar, Apr 12 2021
a(n) = (2/sqrt(5))^n * Product_{j=1..n} i^j*sinh(c*j), where c = arccsch(2) - i*Pi/2. - Peter Luschny, Jul 07 2025

a(0)=1 prepended by Alois P. Heinz, Oct 12 2016

A062381 Let A_n be the n X n matrix defined by A_n[i,j] = 1/F(i+j-1) for 1<=i,j<=n where F(k) is the k-th Fibonacci number (A000045). Then a_n = 1/det(A_n).

Original entry on oeis.org

1, -2, -360, 16848000, 1897448716800000, -3129723891582775706419200000, -541942196790147039091108680776954796441600000, 66373536294235576434745706427960099542896427384297349714149376000000

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 08 2001

In the reference it is proved that not only det(A_n) is a reciprocal of an integer but the inverse matrix (A_n)^(-1) is an integer matrix.

			a(3) = -360 because the matrix is / 1,1,1/2 / 1,1/2, 1/3 / 1/2, 1/3, 1/5 / with determinant -1/360.

Colin Barker, Table of n, a(n) for n = 1..19
T. M. Richardson, The Filbert Matrix, arXiv:math/9905079 [math.RA], 1999.

Cf. A000045, A010048.
Cf. A003266, A152686, A253268.
Cf. A062073, A253267, A253270.

Table[(-1)^Floor[n/2]*Product[Fibonacci[k]^(n-Abs[k-n]),{k,1,2*n-1}],{n,1,10}]/Table[Product[Product[Fibonacci[k],{k,1,m-1}],{m,1,n}],{n,1,10}]^2 (* Alexander Adamchuk, May 18 2006 *)

vector(8, n, 1/matdet(matrix(n, n, i, j, 1/fibonacci(i+j-1)))) \\ Colin Barker, May 01 2015

a(n) = s(n) * f(n) / h(n)^2, where s(n) = (-1)^Floor[n/2], f(n) = Product[Fibonacci[k]^(n-Abs[k-n]),{k,1,2*n-1}], h(n) = Product[Product[Fibonacci[k],{k,1,m-1}],{m,1,n}]. - Alexander Adamchuk, May 18 2006

a(n) ~ (-1)^floor(n/2) * A253270 * ((1+sqrt(5))/2)^(n*(2*n^2+1)/3) / (A253267^2 * 5^(n/2) * A062073^(2*n-2)). - Vaclav Kotesovec, May 01 2015

More terms from Vladeta Jovovic, Jul 11 2001

A067962 a(n) = F(n+2)*(Product_{i=1..n+1} F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number.

Original entry on oeis.org

1, 2, 12, 180, 7200, 748800, 204422400, 145957593600, 272940700032000, 1336044726656640000, 17122749216831498240000, 574502481723130428948480000, 50464872497041500009263431680000, 11605406728144633757130311383449600000

Offset: 0

R. H. Hardin, Feb 02 2002

Number of binary arrangements without adjacent 1's on n X n array connected nw-se.

Kitaev and Mansour give a general formula for the number of binary m X n matrices avoiding certain configurations.

			Neighbors for n=4 (dots represent spaces, circles represent grid points):
O..O..O..O
.\..\..\..
..\..\..\.
O..O..O..O
.\..\..\..
..\..\..\.
O..O..O..O
.\..\..\..
..\..\..\.
O..O..O..O

Reinhard Zumkeller, Table of n, a(n) for n = 0..68
Sergey Kitaev and Toufik Mansour, The problem of the pawns, arXiv:math/0305253 [math.CO], 2003; Annals of Combinatorics 8 (2004) 81-91.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69, 421.

Cf. circle A000204, line A000045, arrays: ne-sw nw-se A067965, e-w ne-sw nw-se A067963, n-s nw-se A067964, e-w n-s nw-se A066864, e-w ne-sw n-s nw-se A063443, n-s A067966, e-w n-s A006506, toruses: bare A002416, ne-sw nw-se A067960, ne-sw n-s nw-se A067959, e-w ne-sw n-s nw-se A067958, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.

Cf. A001654, A003266.

a067962 n = a067962_list !! n
a067962_list = 1 : zipWith (*) a067962_list (drop 2 a001654_list)
-- Reinhard Zumkeller, Sep 24 2015

a:= proc(n) option remember; `if`(n=0, 1, (F->
      F(n+1)*F(n+2)*a(n-1))(combinat[fibonacci]))
    end:
seq(a(n), n=0..14);  # Alois P. Heinz, May 20 2019

Rest[Table[With[{c=Fibonacci[Range[n]]},(Times@@Most[c])^2 Last[c]],{n,15}]] (* Harvey P. Dale, Dec 17 2013 *)

a(n)=fibonacci(n+2)*prod(i=0,n,fibonacci(i+1))^2

a(n) = (F(3) * F(4) * ... * F(n+1))^2 * F(n+2), where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) is asymptotic to C^2*((1+sqrt(5))/2)^((n+2)^2)/(5^(n+3/2)) where C=1.226742010720353244... is the Fibonacci Factorial Constant, see A062073. - Vaclav Kotesovec, Oct 28 2011
a(n) = a(n-1) * A001654(n+1), n > 0. - Reinhard Zumkeller, Sep 24 2015

Edited by Dean Hickerson, Feb 15 2002

Revised by N. J. A. Sloane following comments from Benoit Cloitre, Nov 12 2003

A194157 Product of first n nonzero even-indexed Fibonacci numbers F(2), F(4), F(6), ..., F(2*n).

Original entry on oeis.org

1, 3, 24, 504, 27720, 3991680, 1504863360, 1485300136320, 3838015552250880, 25964175210977203200, 459851507161617245875200, 21322394684069868456741273600, 2588389457883293541569193426124800, 822618641999347403739646931950148812800

Offset: 1

Johannes W. Meijer, Aug 21 2011

The terms of this sequence are Fibonacci double factorial numbers.
a(n) is asymptotic to C2*phi^(n*(n+1))/sqrt(5)^n where phi=(1+sqrt(5))/2 is the golden ratio. For the decimal expansion of C2 see A194159.
Product of first n terms of the binomial transform of the Fibonacci numbers. - Vaclav Kotesovec, Oct 29 2017

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.

Vincenzo Librandi, Table of n, a(n) for n = 1..70
Eric Weisstein, Fibonorial, Mathworld.

Cf. A003266, A062073, A194158, A194160, A194157, A194159, A123029.

[&*[Fibonacci(2*i): i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Sep 15 2016

with(combinat): A194157 :=proc(n): mul(fibonacci(2*i), i=1..n) end: seq(A194157(n), n=1..14);

FoldList[Times, Fibonacci[2 Range[20]]] (* or *)
Table[Round[GoldenRatio^(n(n-1)) QFactorial[n, 1/GoldenRatio^4]], {n, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
Table[Product[Sum[Binomial[m, k]*Fibonacci[k], {k, 1, m}], {m, 1, n}], {n, 1, 12}] (* Vaclav Kotesovec, Oct 29 2017 *)

{a(n) = if( n<0, 0, prod(k=1, n, fibonacci(2*k)))}; /* Michael Somos, Oct 06 2014 */

a(n) = Product_{i=1..n} F(2*i) with F(n) = A000045(n).
a(n) = A123029(2*n).
a(n+1)/a(n) = A001906(n+1).
0 = a(n)*(3*a(n+2)^2 - a(n+1)*a(n+3)) -a(n+1)^2*a(n+2) for all n>=0. - Michael Somos, Oct 06 2014

A082480 a(n) = Product_{k=1..n} (F(k)+1) where F(k) denotes the k-th Fibonacci number.

Original entry on oeis.org

1, 2, 4, 12, 48, 288, 2592, 36288, 798336, 27941760, 1564738560, 140826470400, 20419838208000, 4778242140672000, 1806175529174016000, 1103573248325323776000, 1090330369345419890688000, 1742347930213980985319424000, 4503969399603140847050711040000

Offset: 0

Benoit Cloitre, Apr 27 2003

nonn

Equals row sums (unsigned) of triangle A158472. - Gary W. Adamson, Mar 20 2009

Cf. A000045, A158472. - Gary W. Adamson, Mar 20 2009

with(combinat): a:= n->mul(fibonacci(j)+1, j=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Mar 29 2009

Table[Product[Fibonacci[k]+1,{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 19 2015 *)

PARI
```
a(n)=prod(k=1,n,fibonacci(k)+1)
```

a(n) ~ f * C * ((1+sqrt(5))/2)^(n*(n+1)/2) / 5^(n/2), where C = A062073 = 1.2267420107203532444176302304553616558714096904402504196432973... is the Fibonacci factorial constant and f = Product_{k>=1} (1 + 1/Fibonacci(k)) = 13.150966657784184367612433370626658932190199543164284701354100747157698046... . - Vaclav Kotesovec, Jul 19 2015

Equals the obverse convolution of A000012 and A000045; see A374848. a(n) = (F(n)+1)*a(n-1) for n>=1, where F(n) = A000045(n) = n-th Fibonacci number. - Clark Kimberling, Aug 05 2024

A152686 Partial products of the partial products of the nonzero Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 12, 360, 86400, 269568000, 17662095360000, 39345496591564800000, 4820704671590339051520000000, 52567343238846954009129910272000000000, 82543717140049422917575408530662149324800000000000

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

nonn

Partial products of A003266.

G. C. Greubel, Table of n, a(n) for n = 0..31

Cf. A001924, A003266, A062381, A253267.

Table[Product[Product[Fibonacci[k],{k,1,j}],{j,1,n}],{n,1,12}] (* Vaclav Kotesovec, May 01 2015 *)

a(n) = Product_{i=1..n} A003266(i). - R. J. Mathar, Dec 12 2008

a(n) ~ f * ((1+sqrt(5))/2)^(n*(n+1)*(n+2)/6) * C^n / 5^(n*(n+1)/4), where C = A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant and f = A253267 = 1.096414072507324423110215998844440375945929608777697938465... . - Vaclav Kotesovec, May 01 2015

Edited by R. J. Mathar, Dec 12 2008

a(0)=1 prepended by Alois P. Heinz, Sep 14 2018

A003150 Fibonomial Catalan numbers.

Original entry on oeis.org

1, 1, 3, 20, 364, 17017, 2097018, 674740506, 568965009030, 1255571292290712, 7254987185250544104, 109744478168199574282739, 4346236474244131564253156182, 450625464087974723307205504432150, 122319234225590858340579679211039433810

Offset: 0

N. J. A. Sloane, Henry Gould

			a(5) = F(10)...F(7)/(F(5)...F(1)) = 55*34*21*13/(5*3*2*1*1) = 17017.

H. W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..40
Christian Ballot, Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions, J. Int. Seq., Vol. 21 (2018), Article 18.6.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Henry W. Gould, A new primality criterion of Mann and Shanks and its relation to a theorem of Hermite with extension to Fibonomials, Fib. Quart., 10 (1972), 355-364, 372.
Henry W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
Bruce Sagan, Open Problems for Catalan Number Analogues, January 11, 2015. See FiboCatalan numbers p. 6.
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A000045, A001622, A003267, A010048, A062073.

QBinomial:= func< n, k, q | (&*[( 1-q^(n-j) )/( 1-q^(j+1) ): j in [0..k-1]]) >;
A003150:= func< n | n eq 0 select 1 else Round( ((1+Sqrt(5))/2)^(n^2)*QBinomial( 2*n, n, -2/(3+Sqrt(5)) )/Fibonacci(n+1) ) >;
[A003150(n): n in [0..30]]; // G. C. Greubel, Nov 04 2022

A010048 := proc(n,k) local a,j ; a := 1 ; for j from 0 to k-1 do a := a*combinat[fibonacci](n-j)/combinat[fibonacci](k-j) ; end do: return a; end proc:
A003150 := proc(n) A010048(2*n,n)/combinat[fibonacci](n+1) ; end proc:
seq(A003150(n),n=0..20) ; # R. J. Mathar, Dec 06 2010

f[n_]:= f[n]= Fibonacci[n]; a[n_]:=Product[f[k], {k,n+2,2n}]/Product[f[k], {k,n}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 14 2011 *)
Table[Fibonorial[2 n]/(Fibonorial[n] Fibonorial[n+1]), {n, 0, 20}] (* Since v. 10.0, Vladimir Reshetnikov, May 21 2016 *)
Round@Table[GoldenRatio^(n^2) QBinomial[2 n, n, -1/GoldenRatio^2]/Fibonacci[n + 1], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Sep 25 2016 *)

ft(n) = prod(k=1, n, fibonacci(k)); \\ A003266
fn(n,k) = ft(n)/(ft(k)*ft(n-k)); \\ A010048
a(n) = fn(2*n, n)/fibonacci(n+1); \\ Michel Marcus, Aug 05 2023

def A003150(n): return round( golden_ratio^(n^2)*gaussian_binomial(2*n, n, -1/golden_ratio^2)/fibonacci(n+1) )
[A003150(n) for n in range(30)] # G. C. Greubel, Nov 04 2022

F(2n)*F(2n-1)* ...* F(n+2)/(F(n)*F(n-1)* ... *F(1)) = A010048(2*n,n)/F(n+1), F = Fibonacci numbers.
a(n) ~ sqrt(5) * phi^(n^2-n-1) / C, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and C = A062073 = 1.22674201072035324441763... is the Fibonacci factorial constant. - Vaclav Kotesovec, Apr 10 2015
a(n) = A003267(n)/F(n+1) = A010048(2*n, n)/F(n+1) = phi^(n^2) * C(2*n, n)A001622%20is%20the%20golden%20ratio,%20and%20C(n,%20k)_q%20is%20the%20q-binomial%20coefficient.%20-%20_Vladimir%20Reshetnikov">{-1/phi^2} / F(n+1), where phi = (1+sqrt(5))/2 = A001622 is the golden ratio, and C(n, k)_q is the q-binomial coefficient. - _Vladimir Reshetnikov, Sep 27 2016

A003267 Central Fibonomial coefficients.

Original entry on oeis.org

1, 1, 6, 60, 1820, 136136, 27261234, 14169550626, 19344810307020, 69056421075989160, 645693859487298425256, 15803204856220738696714416, 1012673098498882654470985390406, 169885799961166470686816475170920550, 74614732877610423587753604318734054624100

Offset: 0

N. J. A. Sloane

The largest prime factor of a(n): 1, 1, 3, 5, 13, 17, 89, 233, 233, 1597, 1597, 1597, 28657, 28657, 28657, 514229, 514229, 514229, 514229, 514229, 514229, 514229, 433494437, 433494437, 2971215073, ..., . The union of the above list is: 1, 3, 5, 13, 17, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 14736206161, 46165371073, 92180471494753, 99194853094755497, ... . - Robert G. Wilson v, Dec 04 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.
Phakhinkon Phunphayap, Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes, Ph. D. Thesis, Silpakorn University (Thailand 2021).
Phakhinkon Phunphayap, Prapanpong Pongsriiam, Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients, J. Int. Seq. 21 (2018), #18.3.1.
Eric Weisstein's World of Mathematics, Central Fibonomial Coefficient [From _Eric W. Weisstein_, Dec 08 2009]
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Bisection of A003268. Cf. A008341.

Cf. A001622, A062073, A062381.

with(combinat): a := n -> product(fibonacci(n+k+1), k=0..n-1)/product(fibonacci(k), k=1..n):
seq(a(n), n=0..20);

f[n_] := Product[Fibonacci[n + k + 1]/Fibonacci[k + 1], {k, 0, n - 1}]; Array[f, 14, 0] (* Robert G. Wilson v, Dec 04 2009 *)
Flatten[{1, Table[Round[-(GoldenRatio^(n^2) QPochhammer[(-1)^n GoldenRatio^(-2 n), -GoldenRatio^-2, 1 + n])/((-1 + (-1)^n GoldenRatio^(-2 n)) QPochhammer[ -GoldenRatio^-2, -GoldenRatio^-2, n])], {n,1,15}]}]  (* Vaclav Kotesovec, Apr 10 2015 after Eric W. Weisstein *)
Round@Table[GoldenRatio^(n^2) QBinomial[2 n, n, -1/GoldenRatio^2], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Sep 25 2016 *)

a(n)=prod(k=0,n-1,fibonacci(n+k+1))/prod(k=1,n,fibonacci(k))
for(n=0,14,print1(a(n),","))

For n > 0, a(n) = (-1)^floor(n/2)*det(M(n, -1))/det(M(n, 0)) where M(n, m) is the n X n matrix with coefficient 1/F(i+j+m), i=1..n, j=1..n. - Benoit Cloitre, Jun 05 2004
For n > 0, a(n) = -(GoldenRatio^(n^2) QPochhammer[(-1)^n GoldenRatio^(-2 n), -GoldenRatio^-2, 1 + n])/((-1 + (-1)^n GoldenRatio^(-2 n)) QPochhammer[ -GoldenRatio^-2, -GoldenRatio^-2, n]). - Eric W. Weisstein, Dec 08 2009
a(n) ~ phi^(n^2) / C, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and C = A062073 = 1.22674201072035324441763... is the Fibonacci factorial constant. - Vaclav Kotesovec, Apr 10 2015
a(n) = phi^(n^2) * C(2*n, n)A001622%20is%20the%20golden%20ratio,%20and%20C(n,%20k)_q%20is%20the%20q-binomial%20coefficient.%20-%20_Vladimir%20Reshetnikov">{-1/phi^2}, where phi = (1+sqrt(5))/2 = A001622 is the golden ratio, and C(n, k)_q is the q-binomial coefficient. - _Vladimir Reshetnikov, Sep 26 2016
a(n) = A010048(2*n, n). - Vladimir Reshetnikov, Sep 27 2016

More terms from Sascha Kurz and Rick L. Shepherd, Mar 24 2002
a(1) = 1 added by N. J. A. Sloane, Dec 06 2009
Typo in second formula corrected by Vaclav Kotesovec, Apr 10 2015
Offset corrected from 1 to 0, formulae and programs are updated accordingly by Vladimir Reshetnikov, Sep 27 2016

A056569 Row sums of Fibonomial triangle A010048.

Original entry on oeis.org

1, 2, 3, 6, 14, 42, 158, 756, 4594, 35532, 349428, 4370436, 69532964, 1407280392, 36228710348, 1186337370456, 49415178236344, 2618246576596392, 176462813970065208, 15128228719573952976, 1649746715671916095304

Offset: 0

Wolfdieter Lang, Jul 10 2000

S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

Cf. A010048, A062073, A181926.

Table[Sum[Product[Fibonacci[j],{j,1,n}] / Product[Fibonacci[j],{j,1,k}] / Product[Fibonacci[j],{j,1,n-k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 30 2015 *)
(* Or, since version 10 *) Table[Sum[Fibonorial[n]/Fibonorial[k]/Fibonorial[n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 30 2015 *)
Round@Table[Sum[GoldenRatio^(k(n-k)) QBinomial[n, k, -1/GoldenRatio^2], {k, 0, n}], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Sep 25 2016 *)

ffib(n):=prod(fib(k),k,1,n);
fibonomial(n,k):=ffib(n)/(ffib(k)*ffib(n-k));
makelist(sum(fibonomial(n,k),k,0,n),n,0,30); /* Emanuele Munarini, Apr 02 2012 */

a(n) = Sum_{m=0..n} A010048(n, m), where A010048(n, m) = fibonomial(n, m).
From Vaclav Kotesovec, Apr 30 2015: (Start)
a(n) ~ c * ((1+sqrt(5))/2)^(n^2/4), where
c = EllipticTheta[3,0,1/GoldenRatio] / QPochhammer[-1/GoldenRatio^2] = 2.082828701647012450835512317685120373906427048806222527375... if n is even,
c = EllipticTheta[2,0,1/GoldenRatio] / QPochhammer[-1/GoldenRatio^2] = 2.082828691334156222136965926255238646603356514964103252122... if n is odd.
Or c = Sum_{j} ((1+sqrt(5))/2)^(-(j+(1-(-1)^n)/4)^2) / A062073, where A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant.
(End)

A194158 Product of first n nonzero odd-indexed Fibonacci numbers F(1), ..., F(2*n-1).

Original entry on oeis.org

1, 2, 10, 130, 4420, 393380, 91657540, 55911099400, 89290025741800, 373321597626465800, 4086378207619294646800, 117103340295746126693347600, 8785678105688353155168403690000, 1725665322163094950031867515982420000, 887387152950606153059937200876123854180000

Offset: 1

Johannes W. Meijer, Aug 21 2011

The terms of this sequence are Fibonacci double factorial numbers.

The a(n) is asymptotic to C1*phi^(n*n)/sqrt(5)^n where phi=(1+sqrt(5))/2 is the golden ratio A001622. For the decimal expansion of C1 see A194160.

			G.f. = 1 + x + 2*x^2 + 10*x^3 + 130*x^4 + 4420*x^5 + 393380*x^6 + 91657540*x^7 + ...

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.

G. C. Greubel, Table of n, a(n) for n = 1..69
Eric Weisstein's World of Mathematics, Fibonorial

Cf. A003266, A062073, A194158, A194160, A194157, A194159, A123029.

with(combinat): A194158 :=proc(n): mul(fibonacci(2*i-1), i=1..n) end: seq(A194158(n), n=1..15);

Table[Product[Fibonacci[2*k - 1], {k, 1, n}], {n, 1, 30}] (* G. C. Greubel, Aug 13 2018 *)

{a(n) = if( n<0, 1 / a(-n), prod(k=1, n, fibonacci(2*k - 1)))}; /* Michael Somos, Oct 07 2014 */

a(n) = Product_{i=1..n} F(2*i-1), where F(n) = A000045(n).
a(n) = A123029(2*n-1).
a(n+1)/a(n) = A001519(n+1).
a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 07 2014
a(-n) = 1/a(n) for all n in Z. - Michael Somos, Oct 07 2014
0 = a(n)*(+a(n+1)*a(n+3) - 3*a(n+2)^2) + a(n+2)*(+a(n+1)^2) for all n in Z. - Michael Somos, Oct 07 2014
(F(1) + i)(F(3) + i)...(F(2n+1) + i) = a(n)(1 + F(2n+2)i) and (F(2n+1) + i)(1 + F(2n)i) = F(2n-1)(1 + F(2n+2)i) for all n in Z. - Michael Somos, Sep 16 2023

cp's OEIS Frontend

A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

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A062381 Let A_n be the n X n matrix defined by A_n[i,j] = 1/F(i+j-1) for 1<=i,j<=n where F(k) is the k-th Fibonacci number (A000045). Then a_n = 1/det(A_n).

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A067962 a(n) = F(n+2)*(Product_{i=1..n+1} F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number.

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A194157 Product of first n nonzero even-indexed Fibonacci numbers F(2), F(4), F(6), ..., F(2*n).

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A082480 a(n) = Product_{k=1..n} (F(k)+1) where F(k) denotes the k-th Fibonacci number.

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A152686 Partial products of the partial products of the nonzero Fibonacci numbers.

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