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A005155
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Number of degree sequences of n-node graphs.
(Formerly M1886)
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1
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1, 1, 2, 8, 54, 533, 6944, 111850, 2135740, 47003045, 1168832808, 32363244260, 986532609608, 32810811179569, 1181865951824800, 45823912079507918, 1902469319507438352, 84195282530581058825, 3956365033583165905568
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Given a simple graph, the degree sequence maps each vertex to the valence or degree of that vertex.
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REFERENCES
| R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 (1997) pp. 149-180. See p. 161
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, ``A zonotope associated with graphical degree sequences,'' in Applied Geometry and Discrete Combinatorics. DIMACS Series in Discrete Math., Amer. Math. Soc., Vol. 4, pp. 555-570, 1991.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.16.
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FORMULA
| There is an explicit formula and e.g.f.
E.g.f.: (sqrt((1-LambertW(-x))/(1+LambertW(-x)))-LambertW(-x)/x)*exp(-LambertW(-x)^2/2)/2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 21 2007
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MATHEMATICA
| max = 18; w = ProductLog; f[x_] := (Sqrt[(1 - w[-x])/(1 + w[-x])] - w[-x]/x)*(Exp[-w[-x]^2/2]/ 2); CoefficientList[ Series[f[x], {x, 0, max}], x]*Range[0, max]! (* From Jean-François Alcover, Dec 12 2011, after Vladeta Jovovic *)
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PROG
| (PARI) {a(n)=local(A, B, C); if(n<0, 0, A=sum(k=1, n, k^k*x^k/k!, x*O(x^n)); B=intformal(1+A); C=intformal(1/(1-B)); n!*polcoeff( (1+(1-B)*sqrt(1+2*A))/2*exp(C), n))} /* Michael Somos Aug 19 2005 */
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CROSSREFS
| Cf. A004251 for graphs up to isomorphism.
Sequence in context: A052662 A073564 A199576 * A133316 A005440 A183282
Adjacent sequences: A005152 A005153 A005154 * A005156 A005157 A005158
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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