%I A000629
%S A000629 1,2,6,26,150,1082,9366,94586,1091670,14174522,204495126,3245265146,
%T A000629 56183135190,1053716696762,21282685940886,460566381955706,
%U A000629 10631309363962710,260741534058271802,6771069326513690646
%N A000629 Number of necklaces of sets of labeled beads.
%C A000629 Also the number of logically distinct strings of first order quantifiers 
               in which n variables occur (C. S. Peirce, c. 1903). - Stephen Pollard 
               (spollard(AT)truman.edu), Jun 07 2002
%C A000629 Stirling transform of A052849(n)=[2,4,12,48,240,...] is a(n)=[2,6,26,
               150,1082,..]. - Michael Somos Mar 04 2004
%C A000629 Stirling transform of A000142(n-1)=[1,1,2,6,24,...] is a(n-1)=[1,2,6,
               26,...]. - Michael Somos Mar 04 2004
%C A000629 Stirling transform of (-1)^n*A024167(n-1)=[0,1,-1,5,-14,94,...] is a(n-2)=[0,
               1,2,6,26,...]. - Michael Somos Mar 04 2004
%C A000629 The asymptotic expansion of 2*log(n)-(2^1log(1)+2^2log(2)+...+2^nlog(n))/
               2^n is a(1)/1/n +a(2)/2/n^2 +a(3)/3/n^3 +... - Michael Somos, Aug 
               22 2004
%C A000629 This is the sequence of cumulants of the probability distribution of 
               the number of tails before the first head in a sequence of fair coin 
               tosses. - Michael Hardy (hardy(AT)math.umn.edu), May 01 2005
%C A000629 Appears to be row sums of A154921. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Jan 18 2009]
%D A000629 N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981, p. 36.
%D A000629 Eric Hammer, The Calculations of Peirce's 4.453, Transactions of the 
               Charles S. Peirce Society, Vol. 31 (1995), pp. 829-839.
%D A000629 D. E. Knuth, personal communication.
%D A000629 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 
               174.
%D A000629 Dawidson RAZAFIMAHATOLOTRA, Number of Preorders to Compute Probability 
               of Conflict of an Unstable Effectivity Function, Preprint, Paris 
               School of Economics, University of Paris I, Nov 23 2007.
%D A000629 Charles Sanders Peirce, Collected Papers, eds. C. Hartshorne and P. Weiss, 
               Harvard University Press, Cambridge, Vol. 4, 1933, pp. 364-365.
%H A000629 T. D. Noe, <a href="b000629.txt">Table of n, a(n) for n=0..100</a>
%H A000629 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=99">
               Encyclopedia of Combinatorial Structures 99</a>
%H A000629 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GeometricDistribution.html">Link to a section of The World of Mathematics.</
               a>
%H A000629 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StirlingNumberoftheSecondKind.html">Stirling Number of the Second 
               Kind</a>
%F A000629 For n>0, a(n) = 2*A000670(n).
%F A000629 a(n) = Sum {from k=1 to infinity} k^n/(2^k); a(n) = 1 + Sum {from j=0 
               to n-1} C(n, j) a(j); number of combinations of a Simplex lock having 
               n buttons.
%F A000629 a(n) = round[n!/ln(2)^(n+1)] (just for n <= 15) - Henry Bottomley (se16(AT)btinternet.com), 
               Jul 04 2000
%F A000629 a(n) is asymptotic to n!/log(2)^(n+1). - Benoit Cloitre, Oct 20, 2002
%F A000629 a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*2^k. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Sep 29 2003
%F A000629 E.g.f.: exp(x)/(2-exp(x)) = d/dx log(1/(2-exp(x))).
%F A000629 a(n) = Sum_{k = 1..n} A008292(n, k)*2^k; A008292: triangle of Eulerian 
               numbers . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 05 2004
%F A000629 a(1)=1, a(n) = 2*sum(k! A008277(n-1, k), k=1..n-1) for n>1 or a(n) = 
               sum((k-1)! A008277(n, k), k=1..n) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), 
               Feb 05 2005
%F A000629 a(n)=sum{k=0..n, S2(n+1, k+1)k!} - Paul Barry (pbarry(AT)wit.ie), Apr 
               20 2005
%F A000629 A000629 = binomial transform of this sequence. a(n) = sum of terms in 
               n-th row of A028246 - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 
               30 2005
%F A000629 a(n) = 2*(-1)^n * n!*Laguerre(n,P((.),2)), umbrally, where P(j,t) are 
               the polynomials in A131758. - Tom Copeland (tcjpn(AT)msn.com), Sep 
               28 2007
%e A000629 a(3)=6: the necklace representatives on 1,2,3 are ({123}), ({12},{3}), 
               ({13},{2}), ({23},{1}), ({1},{2},{3}), ({1},{3},{2})
%p A000629 spec := [ B, {B=Cycle(Set(Z,card>=1))}, labeled ]; [seq(combstruct[count](spec, 
               size=n), n=0..20)];
%p A000629 a:=n->add(stirling2(n,k)*(k-1)!,k=1..n); (Zabrocki)
%t A000629 a[ 0 ] = 1; a[ n_ ] := (a[ n ] = 1+Sum[ Binomial[ n, k ] a[ n-k ], {k, 
               1, n} ])
%o A000629 (PARI) a(n)=if(n<0,0,n!*polcoeff(subst((1+y)/(1-y),y,exp(x+x*O(x^n))-1),
               n))
%Y A000629 Same as A076726 except for a(0). Cf. A008965.
%Y A000629 Binomial transform of A000670, also double of A000670 - Joe Keane (jgk(AT)jgk.org)
%Y A000629 A002050(n) = a(n) - 1.
%Y A000629 Cf. A008277.
%Y A000629 Sequence in context: A052859 A103937 A159311 this_sequence A032187 A003659 
               A159602
%Y A000629 Adjacent sequences: A000626 A000627 A000628 this_sequence A000630 A000631 
               A000632
%K A000629 nonn,easy,eigen,nice
%O A000629 0,2
%A A000629 N. J. A. Sloane (njas(AT)research.att.com), D. E. Knuth, Nick Singer 
               (nsinger(AT)eos.hitc.com)