A259314 -id:A259314 - 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

A058694 Partial products p(0)p(1)...*p(n) of partition numbers A000041.

Original entry on oeis.org

1, 1, 2, 6, 30, 210, 2310, 34650, 762300, 22869000, 960498000, 53787888000, 4141667376000, 418308404976000, 56471634671760000, 9939007702229760000, 2295910779215074560000, 681885501426877144320000, 262525918049347700563200000, 128637699844180373275968000000

Offset: 0

N. J. A. Sloane, Dec 30 2000

nonn

a(n) gives the number of partitions P(V(n)) of V(n)=[1,2,3,...,n]. A partition P(V(n)) acts on the components of V(n), i.e., the components of V(n) are partitioned. Therefore a(n) results as the product of the number of partitions P(i) of the component v(i)=i with i=1,...,n. For example, a(3) = 6 because we have 6 list partitions for the list V(n=3)=[1,2,3]: [[1], [1, 1], [2, 1]], [[1], [1, 1], [1, 1, 1]], [[1], [1, 1], [3]], [[1], [2], [2, 1]], [[1], [2], [1, 1, 1]], [[1], [2], [3]]. - Thomas Wieder, Sep 29 2007

Equals the eigensequence of triangle A174712; i.e., Triangle A174712 * A058694 preceded by a 1 shifts left. - Gary W. Adamson, Mar 27 2010

Alois P. Heinz, Table of n, a(n) for n = 0..150
Vaclav Kotesovec, The partition factorial constant and asymptotics of the sequence A058694
Eric Weisstein's MathWorld, Hurwitz Zeta Function

Cf. A000041, A000070, A133018, A259314, A259373, A174712.

a:= proc(n) option remember;
       combinat[numbpart](n)*`if`(n>0, a(n-1), 1)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 21 2012
#
# The constant S in the Maple notation
evalf(Zeta(0, -1/2, 23/24)*sqrt(2/3)*Pi - Zeta(0, 1/2, 23/24)*sqrt(3/2)/Pi+3*(D(GAMMA))(23/24)/(4*Pi^2*GAMMA(23/24)) - (Sum(Zeta(0, j/2, 23/24)*(sqrt(3/2)/Pi)^j/j, j=3..infinity)), 60); # Vaclav Kotesovec, Jun 24 2015

Table[Product[PartitionsP[k], {k, 1, n}], {n, 1, 33}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)

a(n)=prod(k=2,n, numbpart(k)) \\ Charles R Greathouse IV, Jan 14 2017

a(n) ~ C * Product_{k=1..n} (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)), where C = 0.9110167313322499518... is the partition factorial constant A259314. - Vaclav Kotesovec, Jun 24 2015

a(n) ~ C * Gamma(23/24) / (n^(n + 11/24 + 3/(4*Pi^2)) * 2^(2*n) * 3^(n/2) * sqrt(2*Pi)) * exp(Pi*(2*n/3)^(3/2) + n + (11*Pi/(12*sqrt(6)) - sqrt(6)/Pi)*sqrt(n) + S), where C = A259314 and S = Zeta(-1/2, 23/24)*sqrt(2/3)*Pi - Zeta(1/2, 23/24)*sqrt(3/2)/Pi + 3*Gamma'(23/24)/(4*Pi^2*Gamma(23/24)) - Sum_{j>=3} Zeta(j/2, 23/24)*(sqrt(3/2)/Pi)^j/j = -0.02541933397793652709903012019225640813047573968579474..., Zeta is the Hurwitz Zeta Function, in Maple notation Zeta(0,z,v), in Mathematica notation Zeta[z,v], equivalently HurwitzZeta[z,v]. - Vaclav Kotesovec, Jun 24 2015

A259373 a(n) = Product_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 1, 4, 108, 67500, 1134472500, 2009787236572500, 343390991123754492187500, 18843880602308850038793150000000000, 370904101895245095313565571450000000000000000000, 6335115544513765517772271190776403515352524800000000000000000000

Offset: 0

Vaclav Kotesovec, Jun 25 2015

nonn

Cf. A000041, A058694, A133018, A259314, A259405, A265097.

Table[Product[PartitionsP[k]^k,{k,0,n}],{n,0,10}]

a(n) ~ c * Product_{k=1..n} ( (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)) )^k, where c = A259405 = 0.90866166764445489256...

A259405 Decimal expansion of a constant related to A259373.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 26 2015

			0.908661667644454892566581137702159278136942213727370666511234283397226865...

Cf. A000041, A259373, A058694, A259314, A133018.

(* The iteration cycle: *) Do[Print[Product[N[PartitionsP[k]^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)))^k, 150], {k, 1, n}]], {n, 1000, 50000, 1000}]

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

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A062073 Decimal expansion of Fibonacci factorial constant.

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A058694 Partial products p(0)*p(1)*...*p(n) of partition numbers A000041.

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A259373 a(n) = Product_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

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A259405 Decimal expansion of a constant related to A259373.

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A058694 Partial products p(0)p(1)...*p(n) of partition numbers A000041.