A194157 -id:A194157 - 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

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

A194159 Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.

Original entry on oeis.org

8, 3, 2, 8, 8, 3, 2, 4, 4, 0, 3, 3, 9, 1, 2, 9, 8, 2, 4, 5, 0, 2, 5, 6, 6, 4, 3, 1, 3, 6, 1, 4, 2, 2, 9, 4, 2, 2, 7, 3, 2, 1, 5, 1, 9, 9, 4, 0, 9, 0, 5, 0, 3, 2, 4, 5, 1, 5, 4, 2, 2, 4, 0, 8, 9, 2, 5, 7, 6, 0, 6, 4, 8, 3, 9, 8, 5, 4, 5, 9, 9, 3, 4, 0, 8, 9, 1, 1, 6, 9, 2, 5, 6, 6, 8, 0, 5, 5, 8, 1, 8, 2, 1, 4, 9, 5, 1, 3

Offset: 0

Johannes W. Meijer, Aug 21 2011

a(n) = Product_{i=1..n} F(2*i) is asymptotic to C2*phi^(n*(n+1))/sqrt(5)^n where phi = (1+sqrt(5))/2 and F(n) = A000045(n), see A194157. The decimal expansion of the constant C2 is given above.

			C2 = 0.83288324403391298245025664...

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.

Eric Weisstein, Fibonorial Mathworld.

Cf. A003266 and A062073; A194158 and A194160; A194157 and A194159.

Cf. A349272.

digits = 108; NProduct[1 - GoldenRatio^(-4*k), {k, 1, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 200] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013, from 1st formula *)
RealDigits[QPochhammer[1/GoldenRatio^4], 10, 100][[1]] (* Vladimir Reshetnikov, Sep 15 2016 *)

Equals Product_{k>=1} (1-alpha^(2*k)) with alpha = -1/phi^2 and phi = (1+sqrt(5))/2.
Equals Sum_{n>=0} (-1)^binomial(n+1,2)*alpha^A152749(n).
Equals A062073/A194160.

A194160 Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.

Original entry on oeis.org

1, 4, 7, 2, 8, 8, 5, 9, 2, 9, 0, 9, 9, 5, 6, 9, 3, 1, 4, 6, 0, 7, 1, 3, 4, 2, 8, 1, 5, 0, 3, 8, 1, 5, 9, 3, 2, 2, 6, 9, 6, 2, 9, 5, 1, 5, 2, 6, 5, 6, 9, 9, 0, 5, 3, 7, 1, 1, 1, 5, 8, 6, 2, 3, 7, 6, 2, 7, 3, 6, 4, 9, 2, 8, 7, 7, 0, 5, 3, 7, 4, 4, 8, 2, 0, 5, 3, 1, 5, 9, 0, 6, 0, 9, 3, 6, 0

Offset: 1

Johannes W. Meijer, Aug 21 2011

A194158(n) = prod(i=1..n, F(2*i-1) ) is asymptotic to C1*phi^(n*n)/sqrt(5)^n where phi=(1+sqrt(5))/2 and F(n) = A000045(n). The decimal expansion of the constant C1 is given here.

			C1 = 1.4728859290995693146071...

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.

Eric Weisstein, Fibonorial, Mathworld.

Cf. A003266 and A062073; A194158 and A194160; A194157 and A194159.

RealDigits[Product[1-((-1)/GoldenRatio^2)^(2k-1),{k,1000}],10, 100] [[1]] (* Harvey P. Dale, Aug 28 2011 *)
RealDigits[QPochhammer[-GoldenRatio^2, 1/GoldenRatio^4]/(GoldenRatio Sqrt[5]), 10, 100][[1]] (* Vladimir Reshetnikov, Sep 15 2016 *)

C1 = prod(k>=1, 1-alpha^(2*k-1) ) where alpha = (-1/phi^2) and phi = (1+sqrt(5))/2.

C1 = A062073/A194159

A123029 a(2n-1) = Product_{i=1..n} Fibonacci(2i-1) and a(2n) = Product_{i=1..n} Fibonacci(2i).

Original entry on oeis.org

1, 1, 2, 3, 10, 24, 130, 504, 4420, 27720, 393380, 3991680, 91657540, 1504863360, 55911099400, 1485300136320, 89290025741800, 3838015552250880, 373321597626465800, 25964175210977203200, 4086378207619294646800

Offset: 1

Roger L. Bagula, Sep 25 2006

From Johannes W. Meijer, Aug 21 2011: (Start)
An appropriate name for this sequence is Fibonacci double factorial, cf. A006882.
In Parks' article appendix 2, a number triangle T(n,k) with T(n,n) = a(n+1), n>=0, appears if we assume that b(r) = Fibonacci(r); see A103631 and A194005. (End)
The original name of this sequence was: A000045 inside a second linear differential equation recursion: b(n) = b(n-1) + b(n-2) --> Binet(n) of A000045 a(n) = b(n)*a(n-2)/(n*(n-1)).
Bagula also stated that using the solutions to these second order differential equations Markov/ linear recursions can be encoded as analog functions.
Partial products of the odd-indexed Fibonacci numbers interleaved with the partial products of the even-indexed Fibonacci numbers. - Harvey P. Dale, Mar 14 2012

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 24*x^6 + 130*x^7 + 504*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..100
P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov, Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.
Eric Weisstein's World of Mathematics, Fibonorial.

Cf. A000045, A003266, A194157, A194158.

Cf. A006882, A103631, A194005.

function a(n)
  if n lt 2 then return 1;
  else return Fibonacci(n)*a(n-2);
  end if; return a;
end function;
[a(n): n in [1..30]]; // G. C. Greubel, Jul 20 2021

with(combinat): A123029 :=proc(n): if type(n,even) then mul(fibonacci(2*i), i=1..n/2) else mul(fibonacci(2*i-1), i= 1..(n+1)/2) fi: end: seq(A123029(n), n=1..21); # Johannes W. Meijer, Aug 21 2011

a[n_]:= a[n]= If[n<2, 1, Fibonacci[n]*a[n-2]]; Table[a[n], {n, 30}] (* modified by G. C. Greubel, Jul 20 2021 *)
With[{nn=21},Riffle[FoldList[Times,1,Fibonacci[Range[3,nn,2]]],FoldList[ Times,1, Fibonacci[ Range[4,nn+1,2]]]]] (* Harvey P. Dale, Mar 14 2012 *)

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

def a(n): return 1 if (n<2) else fibonacci(n)*a(n-2)
[a(n) for n in (1..30)] # G. C. Greubel, Jul 20 2021

a(n) = n!*c(n) with c(n) = b(n)*c(n-2)/(n*(n-1)), c(0) = 1, c(1) = 1; b(n) = b(n-1) + b(n-2), b(0) = 0, b(1) = 1 and b(n) = F(n) with F(n) = A000045(n).
From Johannes W. Meijer, Aug 21 2011: (Start)
a(n) = F(n)*a(n-2).
a(2*n) = A194157(n) and a(2*n-1) = A194158(n). (End)
a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 07 2014
0 = a(n)*(a(n+2)*a(n+3) - a(n+1)*a(n+4)) + a(n+1)*(+a(n+2)^2) for all n>=0. - Michael Somos, Oct 07 2014

Edited by Johannes W. Meijer, Aug 21 2011

A302999 a(n) = Product_{k=1..n} (Fibonacci(k+2) - 1).

Original entry on oeis.org

1, 1, 2, 8, 56, 672, 13440, 443520, 23950080, 2107607040, 301387806720, 69921971159040, 26290661155799040, 16011012643881615360, 15786858466867272744960, 25195826113120167300956160, 65080818850189392138369761280, 272037822793791659138385602150400

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

nonn

a(n) = determinant of (n + 1) X (n + 1) matrix whose main diagonal consists of the consecutive Fibonacci numbers starting with Fibonacci(2) (1, 2, 3, 5, 8, 13, ...) and all other elements are 1's (see example).

			The matrix begins:
  1  1  1  1  1  1  1  1  ...
  1  2  1  1  1  1  1  1  ...
  1  1  3  1  1  1  1  1  ...
  1  1  1  5  1  1  1  1  ...
  1  1  1  1  8  1  1  1  ...
  1  1  1  1  1 13  1  1  ...
  1  1  1  1  1  1 21  1  ...
  1  1  1  1  1  1  1 34  ...

Alois P. Heinz, Table of n, a(n) for n = 0..97

Cf. A000045, A000071, A003266, A067962, A190535, A194157, A194158, A196870.

b:= proc(n) b(n):= `if`(n<1, [1$2][], (f->
      [f, b(n-1)[2]*(f-1)][])(b(n-1)+b(n-2)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 24 2018

Table[Product[Fibonacci[k + 2] - 1, {k, 1, n}], {n, 0, 17}]
Table[Product[Sum[Fibonacci[j], {j, 1, k}], {k, 1, n}], {n, 0, 17}]
Table[Det[Table[If[i == j, Fibonacci[i + 1], 1], {i, 1, n + 1}, {j, 1, n + 1}]], {n, 0, 17}]

a(n) = Product_{k=1..n} A000071(k+2).
a(n) = Product_{k=1..n} Sum_{j=1..k} A000045(j).
a(n) ~ c * ((1 + sqrt(5))/2)^(n*(n+5)/2) / 5^(n/2), where c = 0.1972502311584232476952451740107000852343536766534965116633336539193... - Vaclav Kotesovec, Apr 17 2018
a(n) = A190535(n-3) for n > 3. - Alois P. Heinz, Apr 25 2018

A349272 a(n) = Product_{k = 1..2n+1} Fibonacci(2k) / Sum_{k = 1..2n+1} Fibonacci(2k).

Original entry on oeis.org

1, 2, 315, 2471040, 918185538816, 16047302734562299200, 13178031727820369629763174400, 508406658175888466343652105865846784000, 921456090985190879093613420564815806955580862464000, 78458394721620642094151397745899367347021362840662985785265356800

Offset: 0

Peter Bala, Nov 12 2021

Let m be an even positive integer. We conjecture that the sequence defined by b_m(n) = Product_{k = 1..2*n+1} Fibonacci(m*k) / Sum_{k = 1..2*n+1} Fibonacci(m*k) is integral. The formula given below proves the conjecture in the present case m = 2. The cases m = 4 and m = 6 of the conjecture can be proved in a similar manner.

More generally, if F(n,x) denotes the n-th Fibonacci polynomial we conjecture that, for each n, the rational function Product_{k = 1..2*n+1} F(m*k,x) / Sum_{k = 1..2*n+1} F(m*k,x) is an integral polynomial.

Wikipedia, Fibonacci polynomials

Cf. A000045, A159951, A175553.

Cf. A003266, A194157.

with(combinat):
seq(mul(fibonacci(2*k), k = 1..2*n+1)/add(fibonacci(2*k), k = 1..2*n+1), n = 0..10);

Table[Product[ Fibonacci[2k],{k,2n+1}]/Sum[Fibonacci[2k],{k,2n+1}],{n,0,9}] (* Stefano Spezia, Nov 13 2021 *)

a(n) = prod(k = 1, 2*n+1, fibonacci(2*k)) / sum(k = 1, 2*n+1, fibonacci(2*k)); \\ Michel Marcus, Nov 12 2021

a(n) = F(2*n+1)/F(2*n+2) * Product_{k = 1..2*n} Fibonacci(2*k), shows a(n) to be integral. Cf. A159951.

a(n) ~ A194159 * phi^(4*n^2 + 2*n - 1) / 5^n, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 31 2023

cp's OEIS Frontend

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

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

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A194159 Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.

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A194160 Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.

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A123029 a(2*n-1) = Product_{i=1..n} Fibonacci(2*i-1) and a(2*n) = Product_{i=1..n} Fibonacci(2*i).

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A302999 a(n) = Product_{k=1..n} (Fibonacci(k+2) - 1).

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A123029 a(2n-1) = Product_{i=1..n} Fibonacci(2i-1) and a(2n) = Product_{i=1..n} Fibonacci(2i).

A349272 a(n) = Product_{k = 1..2n+1} Fibonacci(2k) / Sum_{k = 1..2n+1} Fibonacci(2k).