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 A000629 Number of necklaces of partitions of n+1 labeled beads. 135
 1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646, 185603174638656822266 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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) - (2^1*log(1) + 2^2*log(2) + ... + 2^n*log(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) gives 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 Also the number of disjoint areas of a Venn diagram for n multisets. - Aurelian Radoaca, Jun 27 2016 Also the number of ways of ordering n nonnegative integers, allowing for the possibility of ties, and also comparing the smallest integers with 0. Each comparison with 0 gives two possibilities, x > 0 or x=0. As such, without comparison with 0, we get A000670, the number of ways of ordering n nonnegative integers, allowing for the possibility of ties, or the number of ways n competitors can rank in a competition, allowing for the possibility of ties. For instance, for 2 nonnegative integers x,y, there are the following 6 ways of ordering them: x = y = 0, x = y > 0, x > y = 0, x > y > 0, y > x = 0, y > x > 0. - Aurelian Radoaca, Jul 09 2016 Also the number of ordered set partitions of subsets of {1,...,n}. Also the number of chains of distinct nonempty subsets of {1,...,n}. - Gus Wiseman, Feb 01 2019 Number of combinations of a Simplex lock having n buttons. Row sums of the unsigned cumulant expansion polynomials A127671 and logarithmic polynomials A263634. - Tom Copeland, Jun 04 2021 REFERENCES R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987. 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. 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. (CP 4.453 in the electronic edition of The Collected Papers of Charles Sanders Peirce.) 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. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..424 (terms 0..100 from T. D. Noe) R. Austin, R. K. Guy, & R. Nowakowski, Unpublished notes, 1987 Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays (2011), arXiv preprint arXiv:1105.3043 [math.CO], 2011. J. Int. Seq. 14 (2011) # 11.9.5. Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017. Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018. Paul Barry, Series reversion with Jacobi and Thron continued fractions, arXiv:2107.14278 [math.NT], 2021. Arthur T. Benjamin, Combinatorics and campus security, The UMAP Journal 17.2 (summer 1996), pp. 111-116. Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021. 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. Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020. G. H. E. Duchamp, N. Hoang-Nghia, and A. Tanasa, A word Hopf algebra based on the selection/quotient principle, Séminaire Lotharingien de Combinatoire 68 (2013), Article B68c. Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-Mar 11 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 99 H. K. Kim, D. S. Krotov and J. Y. Lee, Matrices uniquely determined by their lonesums, Linear Algebra and its Applications, 438 (2013) no 7, 3107-3123. Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41. Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161. Konstantin Nestmann and Carsten Timm, Time-convolutionless master equation: Perturbative expansions to arbitrary order and application to quantum dots, arXiv:1903.05132 [cond-mat.mes-hall], 2019. Aurelian Radoaca, Properties of Multisets Compared to Sets J. Randon-Furling and S. Redner, Residence Time Near an Absorbing Set, arXiv:1806.09028 [cond-mat.stat-mech], 2018. John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018. S. L. Snover and N. J. A. Sloane, Correspondence, 1991 J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928. M. Thitsa and W. S. Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, pp. 2786-2813. - From N. J. A. Sloane, Dec 26 2012 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 Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004. FORMULA 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_{k>=1} k^n/2^k. a(n) = 1 + Sum_{j=0..n-1} C(n, j)*a(j). a(n) = round(n!/log(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=1..n-1} k!*A008277(n-1, k) for n>1 or a(n) = Sum_{k=1..n} (k-1)!*A008277(n, k). - Mike Zabrocki, Feb 05 2005 a(n) = Sum_{k=0..n} Stirling2(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)^n*b(n), where 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 G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)^2/(2*x^2*(k+1)^2 - (1 - 2*x - 3*x*k)*(1 - 5*x - 3*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 29 2013 a(n) = log(2)*integral_{x>=0} (ceiling(x))^n * 2^(-x) dx. - Peter Bala, Feb 06 2015 EXAMPLE a(2)=6: the necklace representatives on 1,2,3 are ({123}), ({12},{3}), ({13},{2}), ({23},{1}), ({1},{2},{3}), ({1},{3},{2}) G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 150*x^4 + 1082*x^5 + 9366*x^6 + 94586*x^7 + ... From Gus Wiseman, Feb 01 2019: (Start) The a(3) = 26 ordered set partitions of subsets of {1,2,3} are:   {}  {{1}}  {{2}}  {{3}}  {{12}}    {{13}}    {{23}}    {{123}}                            {{1}{2}}  {{1}{3}}  {{2}{3}}  {{1}{23}}                            {{2}{1}}  {{3}{1}}  {{3}{2}}  {{12}{3}}                                                          {{13}{2}}                                                          {{2}{13}}                                                          {{23}{1}}                                                          {{3}{12}}                                                          {{1}{2}{3}}                                                          {{1}{3}{2}}                                                          {{2}{1}{3}}                                                          {{2}{3}{1}}                                                          {{3}{1}{2}}                                                          {{3}{2}{1}} (End) MAPLE 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); # Mike Zabrocki, Feb 05 2005 MATHEMATICA 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}] (* Robert G. Wilson v, Aug 05 2010 *) a[ n_] := If[ n<0, 0, PolyLog[ -n, 1/2]]; (* 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 *) Join[{1}, Rest[t=30; Range[0, t]! CoefficientList[Series[2/(2 - Exp[x]), {x, 0, t}], x]]] (* Vincenzo Librandi, Jan 02 2016 *) PROG (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 CROSSREFS Same as A076726 except for a(0). Cf. A008965, A052861, A008277. Binomial transform of A000670, also double of A000670. - Joe Keane (jgk(AT)jgk.org) A002050(n) = a(n) - 1. A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012 Row sums of A028246. A diagonal of the triangular array in A241168. Row sums of unsigned A127671 and A263634. Sequence in context: A052859 A103937 A159311 * A185994 A032187 A003659 Adjacent sequences:  A000626 A000627 A000628 * A000630 A000631 A000632 KEYWORD nonn,easy,eigen,nice AUTHOR N. J. A. Sloane, Don Knuth, Nick Singer (nsinger(AT)eos.hitc.com) EXTENSIONS a(19) from Michael Somos, Mar 07 2011 STATUS approved

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Last modified September 27 13:12 EDT 2022. Contains 357062 sequences. (Running on oeis4.)