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A001831
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Number of labeled graded partially ordered sets with n elements of height at most 1.
(Formerly M2956 N1194)
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13
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1, 1, 3, 13, 87, 841, 11643, 227893, 6285807, 243593041, 13262556723, 1014466283293, 109128015915207, 16521353903210521, 3524056001906654763, 1059868947134489801413, 449831067019305308555487
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Labeled posets where for all a,b,c in the set, do not have a<b<c. (Equivalently, labeled posets with no chain of length 3; 3-avoiding posets.) Labeled digraphs where every node has indegree 0 or outdegree 0.
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REFERENCES
| D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Index entries for sequences related to posets
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FORMULA
| Sum((-1)^k*C(n, k)*A047863(k), k=0..n).
a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-1)^(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 04 2003
E.g.f. A(x) = Sum_{n>=0} exp((2^k-1)*x)*x^n/n!. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 27 2007
O.g.f.: Sum_{n>=0} x^n/(1 - (2^n - 1)*x)^(n+1) = Sum_{n>=0} a(n)*x^n. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2009]
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MATHEMATICA
| Join[{1}, Table[Sum[Binomial[n, k](2^k-1)^(n-k), {k, n}], {n, 20}]] (* From Harvey P. Dale, Jan 05 2012 *)
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PROG
| (PARI) {a(n)=n!*polcoeff(sum(k=0, n, exp((2^k-1)*x)*x^k/k!), n)} - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 27 2007
(PARI) {a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k-1)*x +x*O(x^n))^(k+1)), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 15 2009]
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CROSSREFS
| Cf. A002031, A047863, A052332, A001833, A048194.
Row sums of A052296.
Cf. variants: A135753, A135754.
Sequence in context: A167810 A054420 A174278 * A196561 A002725 A097711
Adjacent sequences: A001828 A001829 A001830 * A001832 A001833 A001834
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms, formula and comments from Christian G. Bower (bowerc(AT)usa.net), Dec 15 1999.
Last 4 terms corrected by Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 04 2003
Comments corrected by Joel B. Lewis (jblewis(AT)post.harvard.edu), Mar 28 2011
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