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A000629
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Number of necklaces of partitions of n+1 labeled beads.
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56
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1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646, 185603174638656822266
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OFFSET
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0,2
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COMMENTS
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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
Stirling transform of A052849(n) = [2,4,12,48,240,...] is a(n) = [2,6,26,150,1082,..]. - Michael Somos, Mar 04 2004
Stirling transform of A000142(n-1) = [1,1,2,6,24,...] is a(n-1) = [1,2,6,26,...]. - Michael Somos, Mar 04 2004
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
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
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
Appears to be row sums of A154921. [Mats Granvik, Jan 18 2009]
This is the number of cyclically ordered partitions of n+1 labeled points. The ordered version is A000670. [Michael Somos, Jan 08 2011]
A000670(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 27 2012
Row sums of A154921 as conjectured above by Granvik. a(n) counts the number of outcomes of a race between n horses H1,...,Hn, where if a horse falls it is not ranked. For example, when n = 2 the 6 outcomes are a dead heat, H1 wins H2 second, H2 wins H1 second, H1 wins H2 falls, H2 wins H1 falls or both fall. - Peter Bala, May 15 2012
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REFERENCES
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Mircea I. Cirnu, Determinantal formulas for sum of generalized arithmetic-geometric series, Boletin de la Asociacion Matematica Venezolana, Vol. XVIII, No. 1 (2011), p. 13; http://www.emis.de/journals/BAMV/conten/vol18/BAMV_XVIII-1_p015-028.pdf.
N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981, p. 36.
Eric Hammer, The Calculations of Peirce's 4.453, Transactions of the Charles S. Peirce Society, Vol. 31 (1995), pp. 829-839.
H. K. Kim, D. S. Krotov and J. Y. Lee, Matrices uniquely determined by their lonesums, Linear Algebra and its Applications, 5 Jan, 2013. - From N. J. A. Sloane, Feb 05 2013
D. E. Knuth, personal communication.
J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 174.
Charles Sanders Peirce, Collected Papers, eds. C. Hartshorne and P. Weiss, Harvard University Press, Cambridge, Vol. 4, 1933, pp. 364-365.
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.
M. Thitsa and W. S. Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM J. CONTROL OPTIM, Vol. 50, No. 5, pp. 2786-2813. - From N. J. A. Sloane, Dec 26 2012
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics 11(2) (2004), #R10
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..100
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays (2011), arXiv preprint arXiv:1105.3043
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 99
Eric Weisstein's World of Mathematics, Geometric Distribution
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics 11(2) (2004), #R10
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FORMULA
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a(n) = 2*A000670(n) - 0^n. [Michael Somos, Jan 08 2011]
O.g.f.: Sum_{n>=0} 2^n*n!*x^n / Product_{k=0..n} (1+k*x). [Paul D. Hanna, Jul 20 2011]
E.g.f.: exp(x) / (2 - exp(x)) = d/dx log(1 / (2 - exp(x))).
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.
a(n) = round[n!/ln(2)^(n+1)] (just for n <= 15) - Henry Bottomley, Jul 04 2000
a(n) is asymptotic to n!/log(2)^(n+1). - Benoit Cloitre, Oct 20, 2002
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*2^k. - Vladeta Jovovic, Sep 29 2003
a(n) = Sum_{k = 1..n} A008292(n, k)*2^k; A008292: triangle of Eulerian numbers . - Philippe Deléham, Jun 05 2004
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
a(n)=sum{k=0..n, S2(n+1, k+1)k!} - Paul Barry, Apr 20 2005
A000629 = binomial transform of this sequence. a(n) = sum of terms in n-th row of A028246 - Gary W. Adamson, May 30 2005
a(n) = 2*(-1)^n * n!*Laguerre(n,P((.),2)), umbrally, where P(j,t) are the polynomials in A131758. - Tom Copeland, Sep 28 2007
a(n) = 2^n A(n,1/2); A(n,x) the Eulerian polynomials. [Peter Luschny, Aug 03 2010]
a(n)=(-1)^b(n), b(n)= -2*sum(k=0..n-1, binomial(n,k)*b(k)), b(0)=1. [Vladimir Kruchinin, Jan 29 2011]
Row sums of A028246. Let f(x) = x+x^2. Then a(n+1) = (f(x)*d/dx)^n f(x) evaluated at x = 1. - Peter Bala, Oct 06 2011
O.g.f.: 1+2*x/(U(0)-2*x) where U(k)=1+3*x+3*x*k-2*x*(k+2)*(1+x+x*k)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 14 2011
E.g.f.: exp(x)/(2 - exp(x)) = 2/(2-Q(0))-1;Q(k)=1+x/(2*k+1-x*(2*k+1)/(x+(2*k+2)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 14 2011
G.f.: 1 / (1 - 2*x / (1 - 1*x / (1 - 4*x / (1 - 2*x / (1 - 6*x / ...))))). - Michael Somos, Apr 27 2012
PSUM transform of A162509. BINOMIAL transform is A007047. - Michael Somos, Apr 27 2012
G.f.: 1/G(0) where G(k) = 1 - x*(2*k+2)/( 1 - x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
E.g.f.: 1/E(0) where E(k) = 1 - x/(k+1)/(1 - 1/(1 + 1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 27 2013
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EXAMPLE
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a(2)=6: the necklace representatives on 1,2,3 are ({123}), ({12},{3}), ({13},{2}), ({23},{1}), ({1},{2},{3}), ({1},{3},{2})
1 + 2*x + 6*x^2 + 26*x^3 + 150*x^4 + 1082*x^5 + 9366*x^6 + 94586*x^7 + ...
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MAPLE
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spec := [ B, {B=Cycle(Set(Z, card>=1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..20)];
a:=n->add(stirling2(n, k)*(k-1)!, k=1..n); (Zabrocki)
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MATHEMATICA
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a[ 0 ] = 1; a[ n_ ] := (a[ n ] = 1 + Sum[ Binomial[ n, k ] a[ n-k ], {k, 1, n} ])
Table[ PolyLog[n, 1/2], {n, 0, -18, -1}] (* From Robert G. Wilson v, Aug 05 2010 *)
a[ n_] := If[ n<0, 0, PolyLog[ -n, 1/2]]; Table[a[n], {n, 0, 19}] (* Michael Somos, Mar 07 2011 *)
Table[Sum[(-1)^(n-k) StirlingS2[n, k]k! 2^k, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Oct 21 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n! * polcoeff(subst( (1 + y) / (1 - y), y, exp(x + x * O(x^n)) - 1), n))} /* Michael Somos, Mar 04 2004 */
(PARI) {a(n)=polcoeff(sum(m=0, n, 2^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
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CROSSREFS
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Same as A076726 except for a(0). Cf. A008965.
Binomial transform of A000670, also double of A000670 - Joe Keane (jgk(AT)jgk.org)
A002050(n) = a(n) - 1.
Cf. A008277.
A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012
Sequence in context: A052859 A103937 A159311 * A185994 A032187 A003659
Adjacent sequences: A000626 A000627 A000628 * A000630 A000631 A000632
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KEYWORD
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nonn,easy,eigen,nice
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AUTHOR
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N. J. A. Sloane, D. E. Knuth, Nick Singer (nsinger(AT)eos.hitc.com)
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EXTENSIONS
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a(19) from Michael Somos, Mar 07 2011
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STATUS
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approved
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