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A058694 Partial products p(0)*p(1)*...*p(n) of partition numbers A000041. 13
1, 1, 2, 6, 30, 210, 2310, 34650, 762300, 22869000, 960498000, 53787888000, 4141667376000, 418308404976000, 56471634671760000, 9939007702229760000, 2295910779215074560000, 681885501426877144320000, 262525918049347700563200000, 128637699844180373275968000000 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A118491 A359960 A088257 * A336672 A046853 A136351
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 30 2000
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)