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

%I A003266 M1692 #131 Aug 09 2025 09:41:15
%S A003266 1,1,1,2,6,30,240,3120,65520,2227680,122522400,10904493600,
%T A003266 1570247078400,365867569267200,137932073613734400,
%U A003266 84138564904377984000,83044763560621070208000,132622487406311849122176000,342696507457909818131702784000
%N A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).
%C A003266 Equals right border of unsigned triangle A158472. - _Gary W. Adamson_, Mar 20 2009
%C A003266 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
%C A003266 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,
%C A003266        _
%C A003266      _|_|_
%C A003266    _|_|_|_|_
%C A003266  _|_|_|_|_|_|_
%C A003266 |_|_|_|_|_|_|_|,
%C A003266 and here is one of the a(5)^2 = 900 possible such tilings with our given restrictions:
%C A003266        _
%C A003266      _| |_
%C A003266    _|_|_|_|_
%C A003266  _|___|_|___|_
%C A003266 |_|___|___|_|_|. - _Greg Dresden_ and Jiayi Liu, Aug 23 2024
%D A003266 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second edition, Addison Wesley, p 597
%D A003266 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003266 Alois P. Heinz, <a href="/A003266/b003266.txt">Table of n, a(n) for n = 0..99</a> (terms n = 1..50 from T. D. Noe)
%H A003266 Alfred Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972, p. 74.
%H A003266 Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, <a href="https://arxiv.org/abs/2301.10830">Counting Parking Sequences and Parking Assortments Through Permutations</a>, arXiv:2301.10830 [math.CO], 2023.
%H A003266 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_8">Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
%H A003266 Tipaluck Krityakierne and Thotsaporn Aek Thanatipanonda, <a href="https://doi.org/10.1007/978-3-031-69706-7_11">Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C^2-finite Sequences</a>, in Applied Mathematical Analysis and Computations (SGMC 2021) Springer Proc. Math. Stat., Vol. 471. Springer, Cham, 255-297. See p. 287.
%H A003266 Mathematica Stack Exchange, <a href="http://mathematica.stackexchange.com/questions/19637/">Product of Fibonacci numbers using For/Do/While loops</a>.
%H A003266 Yuri V. Matiyasevich and Richard K. Guy, <a href="http://www.jstor.org/stable/2322322">A new formula for Pi</a>, Amer. Math. Monthly 93 (1986), no. 8, 631-635. Math. Rev. 2000i:11199.
%H A003266 Aidan Sudbury, Arthur Sun, David Treeby, and Edward Wang, <a href="https://arxiv.org/abs/2504.19911">Pick-up Sticks and the Fibonacci Factorial</a>, arXiv:2504.19911 [math.PR], 2025.
%H A003266 Thotsaporn Aek Thanatipanonda and Yi Zhang, <a href="https://arxiv.org/abs/2004.01370">Sequences: Polynomial, C-finite, Holonomic, ...</a>, arXiv:2004.01370 [math.CO], 2020. See pp. 5-6.
%H A003266 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonorial.html">Fibonorial</a>
%H A003266 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A003266 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
%F A003266 a(n+3) = a(n+1)*a(n+2)/a(n) + a(n+2)^2/a(n+1). - _Robert Israel_, May 19 2014
%F A003266 a(0) = 1 by convention since empty products equal 1. - _Michael Somos_, Oct 06 2014
%F A003266 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
%F A003266 Sum_{n>=1} 1/a(n) = A101689. - _Amiram Eldar_, Oct 27 2020
%F A003266 Sum_{n>=1} (-1)^(n+1)/a(n) = A135598. - _Amiram Eldar_, Apr 12 2021
%F A003266 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
%e A003266 a(5) = 30 because the first 5 Fibonacci numbers are 1, 1, 2, 3, 5 and 1 * 1 * 2 * 3 * 5 = 30.
%e A003266 a(6) = 240 because 8 is the sixth Fibonacci number and a(5) * 8 = 240.
%e A003266 a(7) = 3120 because 13 is the seventh Fibonacci number and a(6) * 13 = 3120.
%e A003266 G.f. = 1 + x + x^2 + 2*x^3 + 6*x^4 + 30*x^5 + 240*x^6 + 3120*x^7 + ...
%p A003266 with(combinat): A003266 := n-> mul(fibonacci(i),i=1..n): seq(A003266(n), n=0..20);
%t A003266 Rest[FoldList[Times,1,Fibonacci[Range[20]]]] (* _Harvey P. Dale_, Jul 11 2011 *)
%t A003266 a[ n_] := If[ n < 0, 0, Fibonorial[n]]; (* _Michael Somos_, Oct 23 2017 *)
%t A003266 Table[Round[GoldenRatio^(n(n-1)/2) QFactorial[n, GoldenRatio-2]], {n, 20}] (* _Vladimir Reshetnikov_, Sep 14 2016 *)
%o A003266 (PARI) a(n)=prod(i=1,n,fibonacci(i)) \\ _Charles R Greathouse IV_, Jan 13 2012
%o A003266 (Haskell)
%o A003266 a003266 n = a003266_list !! (n-1)
%o A003266 a003266_list = scanl1 (*) $ tail a000045_list
%o A003266 -- _Reinhard Zumkeller_, Sep 03 2013
%o A003266 (Python)
%o A003266 from itertools import islice
%o A003266 def A003266_gen(): # generator of terms
%o A003266     a,b,c = 1,1,1
%o A003266     while True:
%o A003266         yield c
%o A003266         c *= a
%o A003266         a, b = b, a+b
%o A003266 A003266_list = list(islice(A003266_gen(),20)) # _Chai Wah Wu_, Jan 11 2023
%Y A003266 Cf. A000045, A101689, A135598, A158472.
%Y A003266 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
%Y A003266 Cf. A176343, A238243, A238244.
%K A003266 nonn,easy,nice
%O A003266 0,4
%A A003266 _N. J. A. Sloane_
%E A003266 a(0)=1 prepended by _Alois P. Heinz_, Oct 12 2016