A152686 -id:A152686 - cp's OEIS frontend

A253267 Decimal expansion of a constant related to A152686.

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)).

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

A152687 Partial products operator applied thrice to nonzero Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 24, 8640, 746496000, 201231433728000000, 3554168771933456302080000000000, 139840535301953855934724122328694784000000000000000, 674129921807822677705190163721626103970522196466131271680000000000000000000000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

nonn

Partial products of A152686.

G. C. Greubel, Table of n, a(n) for n = 1..18

Cf. A001924, A003266, A152686.

Nest[FoldList[Times,#]&,Fibonacci[Range[10]],3] (* Harvey P. Dale, Oct 06 2017 *)

a(n) = prod(i=1, n, prod(j=1, i, prod(k=1, j, fibonacci(k)))); \\ Michel Marcus, Sep 15 2018

Edited by R. J. Mathar, Dec 12 2008

One more term (a(10)) from Harvey P. Dale, Oct 06 2017

A152690 Partial sums of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 16, 304, 34864, 24918064, 125436246064, 5056710181206064, 1834938528961266006064, 6658608419043265483506006064, 265790273955000365854215115506006064

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..44

Cf. A152686, A152687, A152688, A152689, A053308, A053309, A053295, A053296.

lst={};p0=1;s0=0;Do[p0*=a[n];s0+=p0;AppendTo[lst,s0],{n,0,4!}];lst
s = 0; lst = {s}; Do[s += BarnesG[n]; AppendTo[lst, s], {n, 2, 13, 1}]; lst (* Zerinvary Lajos, Jul 16 2009 *)
Table[Sum[BarnesG[k+1],{k,1,n}],{n,1,15}] (* Vaclav Kotesovec, Jul 10 2015 *)

G.f.: W(0)/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+1)!/( x*(k+1)! + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
a(n) ~ exp(1/12 - 3*n^2/4) * n^(n^2/2 - 1/12) * (2*Pi)^(n/2) / A, where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015
a(n) = n! * G(n+1) + a(n-1), where G(z) is the Barnes G-function. - Daniel Suteu, Jul 23 2016

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... .

A152688 Partial products of A152687.

Original entry on oeis.org

1, 1, 2, 48, 414720, 309586821120000, 62298599877271470735360000000000, 221419738218975714643056286355472083897548800000000000000000000, 30963454718960054822969246779894642673092903344400531870724683866888280945459200000000000000000000000000000000000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

nonn

Cf. A001924,A003266,A152686,A152687

lst={};p0=p1=p2=p3=1;Do[p0*=a[n];p1*=p0;p2*=p1;p3*=p2;AppendTo[lst,p3],{n,1,2*3!}];lst

Better definition from Omar E. Pol, Aug 06 2009

A265944 Absolute value of the determinant of the matrix whose terms are fibonacci(m+r+s)^(n) with 0 <= r, s <=n, for any m.

Original entry on oeis.org

1, 2, 36, 13824, 324000000, 1209323520000000, 1923567501916569600000000, 3436011282355888738787131392000000000, 18204541483393435808637499286914987185930240000000000, 753091424970084722185225494963366011108371967508480000000000000000000000

Offset: 1

Michel Marcus, Dec 23 2015

nonn

L. Carlitz, Some Determinants Containing Powers of Fibonacci Numbers, The Fibonacci Quarterly, 4.2 (1966), 129-134.
Eric Rowland and Jesus Sistos Barron, Complexity of powers of a constant-recursive sequence, arXiv:2501.14643 [math.NT], 2025. See p. 6.
Aram Tangboonduangjit and Thotsaporn Thanatipanonda, Determinants Containing Powers of Generalized Fibonacci Numbers, arXiv:1512.07025 [math.CO], 2015.

Cf. A001142, A152686.

a(n) = prod(j=0, n, binomial(n, j)) * prod(j=1,n, fibonacci(j)^(n-j+1))^2;

a(n) = (Product_{j=0..n} binomial(n,j)) * (Product_{j=1..n} fibonacci(j)^(n-j+1))^2.

a(n) = A001142(n)*A152686(n)^2.

cp's OEIS Frontend

A253267 Decimal expansion of a constant related to A152686.

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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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A152687 Partial products operator applied thrice to nonzero Fibonacci numbers.

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A152690 Partial sums of superfactorials (A000178).

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

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A152688 Partial products of A152687.

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A265944 Absolute value of the determinant of the matrix whose terms are fibonacci(m+r+s)^(n) with 0 <= r, s <=n, for any m.

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