A256831 -id:A256831 - 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

A256832 Product of first n Pell numbers Pell(1), ... , Pell(n).

Original entry on oeis.org

1, 2, 10, 120, 3480, 243600, 41168400, 16796707200, 16544756592000, 39343431175776000, 225870638380130016000, 3130567047948602021760000, 104751903991408172250111360000, 8462068308233934970708495883520000, 1650314871813323167662424409683488000000

Offset: 1

Vaclav Kotesovec, Apr 10 2015

nonn

Cf. A000129, A256831.

Cf. A003266, A135407, A126772.

Table[Product[Expand[((1+Sqrt[2])^k-(1-Sqrt[2])^k)/(2*Sqrt[2])],{k,1,n}],{n,1,20}]
FoldList[Times,LinearRecurrence[{2,1},{1,2},20]] (* Harvey P. Dale, Oct 07 2015 *)
FoldList[Times, Fibonacci[Range[20], 2]] (* or *)
Table[Round[(1+Sqrt[2])^((n-1)n/2) QFactorial[n, Sqrt[8]-3]], {n, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)

a(n)=my(q=quadgen(8)+1,Q=q); prod(k=2,n, imag(Q*=q)) \\ Charles R Greathouse IV, Feb 14 2022

a(n) = Product_{k=1..n} A000129(k).

a(n) ~ c * ((1+sqrt(2))^(n*(n+1)/2) / 2^(3*n/2)), where c = A256831 = 1.1419825696677912... . - Vaclav Kotesovec, Apr 10 2015

A099929 Central Pellonomial coefficients.

Original entry on oeis.org

1, 2, 30, 2436, 1166438, 3248730940, 52755584809356, 4992850354675749192, 2754130291777980970686150, 8854642279944231931659815098860, 165923943638796574201560736475319416580, 18121679707218614746613513717704194807763644600

Offset: 0

Ralf Stephan, Nov 03 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A000129, A099927, A256831.

p:= proc(n) p(n):= `if`(n<2, n, 2*p(n-1)+p(n-2)) end:
f:= proc(n) f(n):= `if`(n=0, 1, p(n)*f(n-1)) end:
a:= n-> f(2*n)/f(n)^2:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 15 2013

Pell[m_]:=Expand[((1+Sqrt[2])^m-(1-Sqrt[2])^m)/(2*Sqrt[2])]; Table[Product[Pell[k],{k,1,2*n}]/(Product[Pell[k],{k,1,n}])^2,{n,0,20}] (* Vaclav Kotesovec, Apr 10 2015 *)

P=[lucas_number1(n, 2, -1) for n in [0..30]]
[prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]] # Tom Edgar, Apr 10 2015

def a(n): return ((1+sqrt(2))^n^2*q_binomial(2*n, n, -(3-2*sqrt(2)))).simplify_full() # Seiichi Manyama, May 10 2025

a(n) = A099927(2n, n).
a(n) ~ (1+sqrt(2))^(n^2) / c, where c = A256831 = 1.141982569667791206028... . - Vaclav Kotesovec, Apr 10 2015
a(n) = (1 + sqrt(2))^(n^2) * q-binomial(2*n, n, -(sqrt(2) - 1)^2). - Seiichi Manyama, May 10 2025

A259314 Decimal expansion of partition factorial constant.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 24 2015

			0.91101673133224995186154746959468345278073860978008093028132149022759...

Vaclav Kotesovec, The partition factorial constant and asymptotics of the sequence A058694

Cf. A000041, A058694, A062073, A218490, A253924, A256831, A259405.

(* The iteration cycle: *) Do[Print[Product[N[PartitionsP[k]/((E^(Sqrt[2/3]*Sqrt[k-1/24]*Pi) * (1 - Sqrt[3/2]/(Sqrt[k-1/24]*Pi))) / (4*Sqrt[3]*(k-1/24))), 150], {k, 1, n}]], {n, 500, 50000, 500}]

Equals limit n->infinity Product_{k=1..n} p(k) / (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)), where p(k) is the partition function A000041.

A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

Original entry on oeis.org

1, 1, 6, 203, 40222, 46410442, 312163223724, 12237378320283699, 2796071362211148193590, 3723566980632561787914135870, 28901575272390972687956930234335380, 1307480498356321410289575304307661963042110, 344746842780849469098742541704318199701366091840620

Offset: 0

Tom Edgar, Apr 10 2015

nonn

One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from Pell numbers (A000129).

			a(5) = Pell(10)..Pell(7)/Pell(5)..Pell(1) = (2378*985*408*169)/(29*12*5*2*1) = 46410442.
a(3) = A099927(6,3)/Pell(3) = 2436/12 = 203.

Alois P. Heinz, Table of n, a(n) for n = 0..50
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.

Cf. A000129, A099927, A003150, A099929, A256831, A256832.

p:= n-> (<<2|1>, <1|0>>^n)[1, 2]:
a:= n-> mul(p(i), i=n+2..2*n)/mul(p(i), i=1..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2015

Pell[m_]:=Expand[((1+Sqrt[2])^m-(1-Sqrt[2])^m)/(2*Sqrt[2])]; Table[Product[Pell[k],{k,1,2*n}]/(Product[Pell[k],{k,1,n}])^2 / Pell[n+1],{n,0,15}] (* Vaclav Kotesovec, Apr 10 2015 *)

P=[lucas_number1(n, 2, -1) for n in [0..30]]
[1/P[n+1]*prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]]

a(n) = Pell(2n)Pell(2n-1)...Pell(n+2)/Pell(n)Pell(n-1)...Pell(1) = A099927(2*n,n)/Pell(n+1) = A099929(n)/Pell(n+1), where Pell(k) = A000129(k).

a(n) ~ 2^(3/2) * (1+sqrt(2))^(n^2-n-1) / c, where c = A256831 = 1.141982569667791206028... . - Vaclav Kotesovec, Apr 10 2015

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A062073 Decimal expansion of Fibonacci factorial constant.

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A256832 Product of first n Pell numbers Pell(1), ... , Pell(n).

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A099929 Central Pellonomial coefficients.

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A259314 Decimal expansion of partition factorial constant.

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A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

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