The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A058694 Partial products p(0)*p(1)*...*p(n) of partition numbers A000041. 12
 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 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 Cf. A000041, A000070, A133018, A259314, A259373, A174712. Sequence in context: A002110 A118491 A088257 * A046853 A136351 A071290 Adjacent sequences:  A058691 A058692 A058693 * A058695 A058696 A058697 KEYWORD nonn AUTHOR N. J. A. Sloane, Dec 30 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 20 19:53 EST 2020. Contains 331096 sequences. (Running on oeis4.)