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

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

Offset: 0

Author

N. J. A. Sloane, Dec 30 2000

Keywords

• nonn

Comments

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

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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