A062381 -id:A062381 - cp's OEIS frontend

A062073 Decimal expansion of Fibonacci factorial constant.

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

Offset: 1

Jason Earls, Jun 27 2001

The Fibonacci factorial constant is associated with the Fibonacci factorial A003266.

Two closely related constants are A194159 and A194160. [Johannes W. Meijer, Aug 21 2011]

			1.226742010720353244417630230455361655871409690440250419643297301214...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison Wesley, 1990, pp. 478, 571.

Harry J. Smith, Table of n, a(n) for n=1..5000
M. Griffiths, Symmetric rational expressions in the Fibonacci numbers, Fib. Q., 46/47 (2008/2009), 262-267. [_N. J. A. Sloane_, Dec 05 2009]
Simon Plouffe, Fibonacci factorials
Eric Weisstein's World of Mathematics, Fibonacci Factorial Constant

Cf. A003266, A003267, A003268, A056569, A062072, A062381, A135407, A181926.

Cf. A218490, A253924, A256831, A259314, A259405.

RealDigits[N[QPochhammer[-1/GoldenRatio^2], 105]][[1]] (* Alonso del Arte, Dec 20 2010 *)
RealDigits[N[Re[(-1)^(1/24) * GoldenRatio^(1/12) / 2^(1/3) * EllipticThetaPrime[1,0,-I/GoldenRatio]^(1/3)], 120]][[1]] (* Vaclav Kotesovec, Jul 19 2015, after Eric W. Weisstein *)

\p 1300 a=-1/(1/2+sqrt(5)/2)^2; prod(n=1,17000,(1-a^n))

{ default(realprecision, 5080); p=-1/(1/2 + sqrt(5)/2)^2; x=prodinf(k=1, 1-p^k); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062073.txt", n, " ", d)) } \\ Harry J. Smith, Jul 31 2009

C = (1-a)*(1-a^2)*(1-a^3)... 1.2267420... where a = -1/phi^2 and where phi is the Golden ratio = 1/2 + sqrt(5)/2.
C = QPochhammer[ -1/GoldenRatio^2]. [Eric W. Weisstein, Dec 01 2009]
C = A194159 * A194160. [Johannes W. Meijer, Aug 21 2011]
C = exp( Sum_{k>=1} 1/(k*(1-(-(3+sqrt(5))/2)^k)) ). - Vaclav Kotesovec, Jun 08 2013
C = Sum_{k = -inf .. inf} (-1)^((k-1)*k/2) / phi^((3*k-1)*k), where phi = (1 + sqrt(5))/2. - Vladimir Reshetnikov, Sep 20 2016

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

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

A120297 Sum of all matrix elements of n X n matrix M(i,j) = Fibonacci(i+j-1).

Original entry on oeis.org

1, 5, 20, 65, 193, 544, 1489, 4005, 10660, 28193, 74273, 195200, 512257, 1343077, 3519412, 9219105, 24144289, 63224096, 165544721, 433437125, 1134810436, 2971065025, 7778499265, 20364618240, 53315655553, 139582833989

Offset: 1

Alexander Adamchuk, Jul 11 2006

nonn

p^2 divides a(p-1) for p = 5, 11, 19, 29, 31, 41, 59, 61, 71, ... = A038872 (Primes congruent to {0, 1, 4} mod 5), also odd primes p such that where 5 is a square mod p. All squared prime divisors of a(n) also belong to A038872.

			Matrix begins:
  1  1  2  3  5
  1  2  3  5  8
  2  3  5  8 13
  3  5  8 13 21
  5  8 13 21 34

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000045, A038872, A001924, A062381.

Table[Sum[Sum[Fibonacci[i+j-1],{i,1,n}],{j,1,n}],{n,1,50}]

a(n) = Sum_{j=1..n} Sum_{i=1..n} Fibonacci(i+j-1).
a(n) = Fibonacci(2*n+3) - 2*Fibonacci(n+3) + 2. - Vladeta Jovovic, Jul 21 2006
G.f.: (1 - x^3 + 2*x^2)/((x-1)*(x^2 + x - 1)*(x^2 - 3*x + 1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009

A253267 Decimal expansion of a constant related to A152686.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 01 2015

			1.0964140725073244231102159988444403759459296087776979384650568137801405219...

Cf. A062073, A062381, A152686.

(* Iterations: *) Do[Print[N[Product[Product[Fibonacci[k],{k,1,j}],{j,1,n}] / (GoldenRatio^(n*(n+1)*(n+2)/6) * QPochhammer[-1/GoldenRatio^2]^n / 5^(n*(n+1)/4)),120]],{n,100,1000,100}]

Equals limit n->infinity A152686(n) / (((1+sqrt(5))/2)^(n*(n+1)*(n+2)/6) * A062073^n / 5^(n*(n+1)/4)).

A253270 Decimal expansion of a constant related to A253268.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 01 2015

			1.1188674438617789501735355690557711005492672642422237428157122531318...

Cf. A062381, A253268.

(* Iterations: *) Do[Print[N[Product[Fibonacci[k]^k,{k,1,n}] / (GoldenRatio^(n*(n+1)*(2*n+1)/6) / 5^(n*(n+1)/4)),120]],{n,100,1000,100}]

Equals limit n->infinity A253268(n) / (((1+sqrt(5))/2)^(n*(n+1)*(2*n+1)/6) / 5^(n*(n+1)/4)).

A253268 Product_{k=1..n} Fibonacci(k)^k.

Original entry on oeis.org

1, 1, 8, 648, 2025000, 530841600000, 33309523161907200000, 1259861409934788058133299200000, 76494996113757662751632456649087438028800000, 19375812441937279261781767910157290423127539712000000000000000

Offset: 1

Vaclav Kotesovec, May 01 2015

Cf. A000045, A003266, A062381, A152686, A253270.

Table[Product[Fibonacci[k]^k,{k,1,n}],{n,1,10}]
FoldList[Times,Table[Fibonacci[n]^n,{n,10}]] (* Harvey P. Dale, Oct 17 2019 *)

a(n) ~ c * ((1+sqrt(5))/2)^(n*(n+1)*(2*n+1)/6) / 5^(n*(n+1)/4), where c = A253270 = 1.1188674438617789501735355690557711005492672642422237428157122531318... .

A103207 a(n)=(-1)^floor(n/2)/det(M_n) where M_n is the n X n matrix of terms 1/(i+j)! i and j ranging from 1 to n.

Original entry on oeis.org

1, 2, 144, 1036800, 1463132160000, 668986161758208000000, 148045794139338685651353600000000, 22147346968743318573346465338485637120000000000

Offset: 0

Benoit Cloitre, Mar 19 2005

nonn

Cf. A062381.

seq(mul(mul(k+j,j=1..n), k=0..n), n=0..7); # Zerinvary Lajos, Jun 01 2007

Table[1/2^n*(Product[(2*k)!/k!,{k,1,n}])^2,{n,0,10}] (* Vaclav Kotesovec, May 01 2015 *)
Table[2^(2*n^2 + n - 1/12) * Glaisher^3 * BarnesG[n+3/2]^2 / (E^(1/4) * Pi^(n+1/2)),{n,0,10}] (* Vaclav Kotesovec, May 01 2015 *)

PARI
```
a(n)=(1/2^n)*prod(k=1,n,(2*k)!/k!)^2
```

a(n)=(1/2^n)*{prod(k=1, n, (2*k)!/k!)}^2.

a(n) ~ A * 2^(2*n^2 + 2*n + 5/12) * n^(n^2 + n + 1/12) / exp(3*n^2/2 + n + 1/12), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, May 01 2015

A119997 Sum of all matrix elements of n X n matrix M[i,j] = (-1)^(i+j)*Fibonacci[i+j-1].

Original entry on oeis.org

1, 1, 4, 5, 17, 32, 97, 225, 628, 1573, 4225, 10880, 28769, 74849, 196708, 513765, 1347025, 3523360, 9229441, 24154625, 63251156, 165571781, 433507969, 1134881280, 2971250497, 7778684737, 20365103812, 53316141125, 139584105233, 365434903328, 956722661665

Offset: 1

Alexander Adamchuk, Aug 03 2006

Prime p divides a(p-1) for p={5,11,19,29,31,41,59,61,71,...} = A038872[n] Primes congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square mod p. p^2 divides a(p-1) for prime p={11,19,29,31,41,59,61,71,...} = A045468[n] Primes congruent to {1, 4} mod 5. Square prime divisors of a(n) up to n=50 are{2,3,5,7,11,13,19,23,29,31,41,47,89,101,139,151,199,211,461,521,3571,9349}, It appears that square prime divisors of a(n) belong to A061446[n] Primitive part of Fibonacci(n), A001578[n] Smallest primitive prime factor of Fibonacci number F(n) and A072183[n] Sequence arising from factorization of the Fibonacci numbers. Sum[Sum[Fibonacci[i+j-1],{i,1,n}],{j,1,n}] = A120297[n]. Sum[Sum[i+j-1,{i,1,n}],{j,1,n}] = n^3. Sum[Sum[(-1)^(i+j)*(i+j-1),{i,1,n}],{j,1,n}] = n for odd n and = 0 for even n.

			Matrix begins:
1 -1 2 -3 5
-1 2 -3 5 -8
2 -3 5 -8 13
-3 5 -8 13 -21
5 -8 13 -21 34

Colin Barker, Table of n, a(n) for n = 1..1000 [Terms up to n=200 from _Vincenzo Librandi_]
Index entries for linear recurrences with constant coefficients, signature (3,1,-7,5,-1).

Cf. A120297, A000045, A038872, A001924, A062381, A038872, A045468, A061446, A001578, A072183.

Table[Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}],{n,1,50}]

a(n) = sum(i=1, n, sum(j=1, n, (-1)^(i+j)*fibonacci(i+j-1))) \\ Colin Barker, Mar 26 2015

Vec(-x*(x^3+2*x-1)/((x-1)*(x^2-3*x+1)*(x^2-x-1)) + O(x^100)) \\ Colin Barker, Mar 26 2015

a(n) = Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}].
a(n) = 3*a(n-1)+a(n-2)-7*a(n-3)+5*a(n-4)-a(n-5) for n>5. - Colin Barker, Mar 26 2015
G.f.: -x*(x^3+2*x-1) / ((x-1)*(x^2-3*x+1)*(x^2-x-1)). - Colin Barker, Mar 26 2015

A296056 Determinant of the inverse of the matrix A_n, where A_n is the n X n matrix defined by A_n[i,j] = 1/C(i+j-2) for 1 <= i,j <= n, and C(k) is the k-th Catalan number (A000108).

Original entry on oeis.org

1, -2, -1400, -679140000, -122489812645200000, -6931927717187904217987200000, -114287375178291587421201860354580633600000, -527655997339226839875614785993553970321322576128000000000, -666218073328701414704702576237379472614149140939534461737723520000000000000

Offset: 1

Tom Richardson, Dec 03 2017

sign

It is conjectured that a(n) is an integer for all n.

The contributor suggests the name "Catbert matrix" for the matrix A_n, based on its similarity to the Hilbert matrix and its relation to the Catalan numbers.

Tom Richardson, Table of n, a(n) for n = 1..29
Tom Richardson, Table of n, a(n) for n = 1..100
Thomas M. Richardson, Catalan Numbers and Jacobi Polynomials, arXiv:2005.08939 [math.CO], 2020.

Cf. A000108, A005249, A062381.

a[n_] := 1/Det@ Table[ 1/CatalanNumber[i + j -2], {i, n}, {j, n}]; Array[a, 9] (* Robert G. Wilson v, Jan 05 2018 *)
Table[Product[4^(2*k + 1) * (4*k - 1)/6 * Binomial[2*k - 3/2, k] * Binomial[2*k - 3/2, k + 1], {k, 0, n - 1}], {n, 1, 10}] (* Vaclav Kotesovec, May 19 2020 *)

a(n) = 1/matdet(matrix(n,n,i,j,(i+j-1)/binomial(2*i+2*j-4,i+j-2)))

a(n) ~ -c * 16^(n*(n-1)) / (3^n * Pi^n * n^(27/8)), where c = 3*A^(3/2) / (2^(7/6) * exp(1/8) * sqrt(Pi)) = 0.9662886794923866798595701447717791386557874..., where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 19 2020

cp's OEIS Frontend

A062073 Decimal expansion of Fibonacci factorial constant.

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

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A003267 Central Fibonomial coefficients.

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A120297 Sum of all matrix elements of n X n matrix M(i,j) = Fibonacci(i+j-1).

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A253267 Decimal expansion of a constant related to A152686.

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A253270 Decimal expansion of a constant related to A253268.

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A253268 Product_{k=1..n} Fibonacci(k)^k.

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A103207 a(n)=(-1)^floor(n/2)/det(M_n) where M_n is the n X n matrix of terms 1/(i+j)! i and j ranging from 1 to n.

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