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%I #59 Jan 14 2017 18:15:29
%S 1,1,2,6,30,210,2310,34650,762300,22869000,960498000,53787888000,
%T 4141667376000,418308404976000,56471634671760000,9939007702229760000,
%U 2295910779215074560000,681885501426877144320000,262525918049347700563200000,128637699844180373275968000000
%N Partial products p(0)*p(1)*...*p(n) of partition numbers A000041.
%C 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
%C Equals the eigensequence of triangle A174712; i.e., Triangle A174712 * A058694 preceded by a 1 shifts left. - _Gary W. Adamson_, Mar 27 2010
%H Alois P. Heinz, <a href="/A058694/b058694.txt">Table of n, a(n) for n = 0..150</a>
%H Vaclav Kotesovec, <a href="/A058694/a058694_2.pdf">The partition factorial constant and asymptotics of the sequence A058694</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>
%F 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
%F 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
%p a:= proc(n) option remember;
%p combinat[numbpart](n)*`if`(n>0, a(n-1), 1)
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Apr 21 2012
%p #
%p # The constant S in the Maple notation
%p 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
%t Table[Product[PartitionsP[k], {k, 1, n}], {n, 1, 33}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)
%o (PARI) a(n)=prod(k=2,n, numbpart(k)) \\ _Charles R Greathouse IV_, Jan 14 2017
%Y Cf. A000041, A000070, A133018, A259314, A259373, A174712.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Dec 30 2000
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