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%I A000629 #304 Jun 04 2024 16:55:00
%S A000629 1,2,6,26,150,1082,9366,94586,1091670,14174522,204495126,3245265146,
%T A000629 56183135190,1053716696762,21282685940886,460566381955706,
%U A000629 10631309363962710,260741534058271802,6771069326513690646,185603174638656822266,5355375592488768406230
%N A000629 Number of necklaces of partitions of n+1 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) - (2^1*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
%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. - _Mats Granvik_, Jan 18 2009
%C A000629 This is the number of cyclically ordered partitions of n+1 labeled points. The ordered version is A000670. - _Michael Somos_, Jan 08 2011
%C A000629 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
%C A000629 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
%C A000629 Also the number of disjoint areas of a Venn diagram for n multisets. - _Aurelian Radoaca_, Jun 27 2016
%C A000629 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
%C A000629 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
%C A000629 Number of combinations of a Simplex lock having n buttons.
%C A000629 Row sums of the unsigned cumulant expansion polynomials A127671 and logarithmic polynomials A263634. - _Tom Copeland_, Jun 04 2021
%C A000629 Also the number of vertices in the axis-aligned polytope consisting of all vectors x in R^n where, for all k in {1,...,n}, the k-th smallest coordinate of x lies in the interval [0, k]. - _Adam P. Goucher_, Jan 18 2023
%C A000629 Number of idempotent Boolean relation matrices whose complement is also idempotent. See Rosenblatt link. - _Geoffrey Critzer_, Feb 26 2023
%D A000629 R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
%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 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.)
%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.
%H A000629 Seiichi Manyama, Table of n, a(n) for n = 0..424 (terms 0..100 from T. D. Noe)
%H A000629 R. Austin, R. K. Guy, & R. Nowakowski, Unpublished notes, 1987
%H A000629 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.
%H A000629 Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
%H A000629 Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
%H A000629 Paul Barry, Series reversion with Jacobi and Thron continued fractions, arXiv:2107.14278 [math.NT], 2021.
%H A000629 Arthur T. Benjamin, Combinatorics and campus security, The UMAP Journal 17.2 (summer 1996), pp. 111-116.
%H A000629 Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 27, 29.
%H A000629 Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
%H A000629 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.
%H A000629 Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
%H A000629 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.
%H A000629 Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019.
%H A000629 Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
%H A000629 W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 Mar 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
%H A000629 Robin Houston, Adam P. Goucher, and Nathaniel Johnston, A New Formula for the Determinant and Bounds on Its Tensor and Waring Ranks, arXiv:2301.06586 [math.CO], 2023.
%H A000629 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 99
%H A000629 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.
%H A000629 Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
%H A000629 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.
%H A000629 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.
%H A000629 Aurelian Radoaca, Properties of Multisets Compared to Sets
%H A000629 J. Randon-Furling and S. Redner, Residence Time Near an Absorbing Set, arXiv:1806.09028 [cond-mat.stat-mech], 2018.
%H A000629 D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
%H A000629 John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.
%H A000629 S. L. Snover and N. J. A. Sloane, Correspondence, 1991
%H A000629 J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.
%H A000629 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
%H A000629 Eric Weisstein's World of Mathematics, Geometric Distribution
%H A000629 Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind
%H A000629 Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics 11(2) (2004), #R10
%H A000629 Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.
%F A000629 a(n) = 2*A000670(n) - 0^n. - _Michael Somos_, Jan 08 2011
%F A000629 O.g.f.: Sum_{n>=0} 2^n*n!*x^n / Product_{k=0..n} (1+k*x). - _Paul D. Hanna_, Jul 20 2011
%F A000629 E.g.f.: exp(x) / (2 - exp(x)) = d/dx log(1 / (2 - exp(x))).
%F A000629 a(n) = Sum_{k>=1} k^n/2^k.
%F A000629 a(n) = 1 + Sum_{j=0..n-1} C(n, j)*a(j).
%F A000629 a(n) = round(n!/log(2)^(n+1)) (just for n <= 15). - _Henry Bottomley_, 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_, Sep 29 2003
%F A000629 a(n) = Sum_{k=1..n} A008292(n, k)*2^k; A008292: triangle of Eulerian numbers. - _Philippe Deléham_, Jun 05 2004
%F A000629 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
%F A000629 a(n) = Sum_{k=0..n} Stirling2(n+1, k+1)*k!. - _Paul Barry_, 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_, 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_, Sep 28 2007
%F A000629 a(n) = 2^n*A(n,1/2); A(n,x) the Eulerian polynomials. - _Peter Luschny_, Aug 03 2010
%F A000629 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
%F A000629 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
%F A000629 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
%F A000629 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
%F A000629 G.f.: 1 / (1 - 2*x / (1 - 1*x / (1 - 4*x / (1 - 2*x / (1 - 6*x / ...))))). - _Michael Somos_, Apr 27 2012
%F A000629 PSUM transform of A162509. BINOMIAL transform is A007047. - _Michael Somos_, Apr 27 2012
%F A000629 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
%F A000629 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
%F A000629 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
%F A000629 a(n) = log(2)*integral_{x>=0} (ceiling(x))^n * 2^(-x) dx. - _Peter Bala_, Feb 06 2015
%e A000629 a(2)=6: the necklace representatives on 1,2,3 are ({123}), ({12},{3}), ({13},{2}), ({23},{1}), ({1},{2},{3}), ({1},{3},{2})
%e A000629 G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 150*x^4 + 1082*x^5 + 9366*x^6 + 94586*x^7 + ...
%e A000629 From _Gus Wiseman_, Feb 01 2019: (Start)
%e A000629 The a(3) = 26 ordered set partitions of subsets of {1,2,3} are:
%e A000629 {} {{1}} {{2}} {{3}} {{12}} {{13}} {{23}} {{123}}
%e A000629 {{1}{2}} {{1}{3}} {{2}{3}} {{1}{23}}
%e A000629 {{2}{1}} {{3}{1}} {{3}{2}} {{12}{3}}
%e A000629 {{13}{2}}
%e A000629 {{2}{13}}
%e A000629 {{23}{1}}
%e A000629 {{3}{12}}
%e A000629 {{1}{2}{3}}
%e A000629 {{1}{3}{2}}
%e A000629 {{2}{1}{3}}
%e A000629 {{2}{3}{1}}
%e A000629 {{3}{1}{2}}
%e A000629 {{3}{2}{1}}
%e A000629 (End)
%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+1,k)*(k-1)!,k=1..n+1); # _Mike Zabrocki_, Feb 05 2005
%t A000629 a[ 0 ] = 1; a[ n_ ] := (a[ n ] = 1 + Sum[ Binomial[ n, k ] a[ n-k ], {k, 1, n} ])
%t A000629 Table[ PolyLog[n, 1/2], {n, 0, -18, -1}] (* _Robert G. Wilson v_, Aug 05 2010 *)
%t A000629 a[ n_] := If[ n<0, 0, PolyLog[ -n, 1/2]]; (* _Michael Somos_, Mar 07 2011 *)
%t A000629 Table[Sum[(-1)^(n-k) StirlingS2[n,k]k! 2^k,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Oct 21 2011 *)
%t A000629 Join[{1}, Rest[t=30; Range[0, t]! CoefficientList[Series[2/(2 - Exp[x]), {x, 0, t}], x]]] (* _Vincenzo Librandi_, Jan 02 2016 *)
%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))} /* _Michael Somos_, Mar 04 2004 */
%o A000629 (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
%o A000629 (Python)
%o A000629 from math import comb
%o A000629 from functools import lru_cache
%o A000629 @lru_cache(maxsize=None)
%o A000629 def A000629(n): return 1+sum(comb(n,j)*A000629(j) for j in range(n)) if n else 1 # _Chai Wah Wu_, Sep 25 2023
%Y A000629 Same as A076726 except for a(0). Cf. A008965, A052861, A008277.
%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 A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - _N. J. A. Sloane_, Jul 04 2012
%Y A000629 Row sums of A028246.
%Y A000629 A diagonal of the triangular array in A241168.
%Y A000629 Row sums of unsigned A127671 and A263634.
%K A000629 nonn,easy,eigen,nice
%O A000629 0,2
%A A000629 _N. J. A. Sloane_, _Don Knuth_, Nick Singer (nsinger(AT)eos.hitc.com)
%E A000629 a(19) from _Michael Somos_, Mar 07 2011
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