A259373 -id:A259373 - cp's OEIS frontend

A259405 Decimal expansion of a constant related to A259373.

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.

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

A133018 Partition number of n, raised to power n.

Original entry on oeis.org

1, 1, 4, 27, 625, 16807, 1771561, 170859375, 54875873536, 19683000000000, 17080198121677824, 16985107389382393856, 43439888521963583647921, 113809328043328941786781301, 667840509835890864312744140625, 4816039244598889571670527496421376

Offset: 0

Omar E. Pol, Oct 31 2007

nonn

			a(6)=1771561 because the partition number of 6 is 11 and 11^6=1771561.

Cf. A000312, A058694, A062457, A133032, A259373, A265094. Partition numbers: A000041.

A000041 := proc(n) combinat[numbpart](n) ; end: A133018 := proc(n) A000041(n)^n ; end: seq(A133018(n),n=0..18) ; # R. J. Mathar, Jan 13 2008

Table[PartitionsP[n]^n,{n,0,15}] (* James C. McMahon, Mar 10 2025 *)

a(n) = A000041(n)^n.

a(n) ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n). - Vaclav Kotesovec, Jun 23 2015

More terms from R. J. Mathar, Jan 13 2008

a(15) from James C. McMahon, Mar 10 2025

A259436 a(n) = Sum_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 6, 33, 658, 17465, 1789026, 172648401, 55048521937, 19738048521937, 17099936170199761, 17002207325552593617, 43456890729289136241538, 113852784934058230923022839, 667954362620824922543667163464, 4816707198961510396593071163584840

Offset: 0

Vaclav Kotesovec, Jun 27 2015

nonn

Cf. A000041, A133018, A259373, A259399, A259437, A259438, A265095.

Table[Sum[PartitionsP[k]^k,{k,0,n}],{n,0,15}]

a(n) ~ p(n)^n ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n).

A265097 a(n) = Product_{k=0..n} q(k)^k, where q(k) = partition numbers into distinct parts (A000009).

Original entry on oeis.org

1, 1, 1, 8, 128, 31104, 127401984, 9953280000000, 16717688340480000000, 2243810146471316029440000000, 22438101464713160294400000000000000000, 16671697210628551555613518410547200000000000000000, 2163091659500402360172559530668851200000000000000000000000000000

Offset: 0

Vaclav Kotesovec, Dec 01 2015

nonn

Cf. A058694, A152827, A259373.

Table[Product[PartitionsQ[k]^k, {k,0,n}], {n,0,12}]

cp's OEIS Frontend

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.

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A133018 Partition number of n, raised to power n.

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A259436 a(n) = Sum_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

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A265097 a(n) = Product_{k=0..n} q(k)^k, where q(k) = partition numbers into distinct parts (A000009).

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