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A079144
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Number of labeled interval orders on n elements: (2+2)-free posets.
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7
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1, 3, 19, 207, 3451, 81663, 2602699, 107477247, 5581680571, 356046745023, 27365431508779, 2494237642655487, 266005087863259291, 32815976815540917183, 4636895313201764853259, 743988605732990946684927
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| GRAHAM BRIGHTWELL AND MITCHEL T. KELLER, ASYMPTOTIC ENUMERATION OF LABELLED INTERVAL ORDERS, arXiv 1111.6766
Anders Claesson, Mark Dukes and Martina Kubitzke, Partition and composition matrices, arXiv:1006.1312.
D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40(5) (2001), 945-960.
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FORMULA
| (1/(24^n))*sum(binomial(n, k)*A002439(k), k=0..n). Zagier 2001, p. 954.
G.f.: Sum(Product(1-exp(-k*x),k = 1 .. n),n = 0 .. infinity). a(n) = Sum_{k=0..n} k!*Stirling2(n,k)*A138265(k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 11 2008
Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 19 2009: (Start)
Conjectural form for the o.g.f. as a continued fraction:
1/(1-x/(1-2*x/(1-5*x/(1-7*x/(1-12*x/(1-15*x/(1- ...))))))) = 1 + x + 3*x^2 + 19*x^3 + 207*x^4 + ..., where the sequence [1,2,5,7,12,15,..] is the sequence of generalised pentagonal numbers A001318. Compare with the continued fraction form of the o.g.f. of A002105. (End)
E.g.f.: E(x)=1+(exp(x)-1)/(G(0)+1-exp(x)) ; G(k)= 2*exp(x*(k+1))-1-exp(x*(k+1))*(exp(x*(k+2))-1)/G(k+1); (continued fraction Euler's kind, 1-step). - Sergei N. Gladkovskii, Jan 06 2012
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CROSSREFS
| Cf. A022493 (unlabeled interval orders), A002439 (Glaisher's T numbers).
Sequence in context: A182956 A052886 A180563 * A049056 A204262 A165356
Adjacent sequences: A079141 A079142 A079143 * A079145 A079146 A079147
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KEYWORD
| nonn,easy
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AUTHOR
| Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002
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