OFFSET
0,3
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
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
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 *)
PROG
(PARI) a(n)=prod(k=2, n, numbpart(k)) \\ Charles R Greathouse IV, Jan 14 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 30 2000
STATUS
approved