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A058694
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Partial products p(0)*p(1)*...*p(n) of partition numbers A000041.
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13
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1, 1, 2, 6, 30, 210, 2310, 34650, 762300, 22869000, 960498000, 53787888000, 4141667376000, 418308404976000, 56471634671760000, 9939007702229760000, 2295910779215074560000, 681885501426877144320000, 262525918049347700563200000, 128637699844180373275968000000
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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a:= proc(n) option remember;
combinat[numbpart](n)*`if`(n>0, a(n-1), 1)
end:
#
# 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
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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