login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000629 Number of necklaces of partitions of n+1 labeled beads. 59
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) - (2^1log(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

REFERENCES

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.

G. H. E Duchamp, N. Hoang-Nghia, A. Tanasa, A word Hopf algebra based on the selection/quotient principle, Séminaire Lotharingien de Combinatoire 68 (2013), Article B68c.

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 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.

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

LINKS

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

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 {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, 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

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

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 + ...

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); (Zabrocki)

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}] (* From 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 *)

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.

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

Row sums of A028246.

Sequence in context: A052859 A103937 A159311 * A185994 A032187 A003659

Adjacent sequences:  A000626 A000627 A000628 * A000630 A000631 A000632

KEYWORD

nonn,easy,eigen,nice,changed

AUTHOR

N. J. A. Sloane, D. E. Knuth, Nick Singer (nsinger(AT)eos.hitc.com)

EXTENSIONS

a(19) from Michael Somos, Mar 07 2011

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified April 18 18:41 EDT 2014. Contains 240732 sequences.