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A104209
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Number of labeled directed multigraphs with n arrows and no vertex of degree 0.
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8
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1, 3, 39, 819, 23949, 898947, 41212155, 2232057171, 139455901101, 9873341493231, 781184921112075, 68309191570851759, 6541702440222052137, 680922615974259589527, 76544749927261960908807
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| These are the dimensions of the homogeneous components of a commutative graded Hopf algebra generalizing quasi-symmetric functions.
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REFERENCES
| J.-C. Novelli, J.-Y. Thibon and N. M. Thiery, Algebres de Hopf de graphes, C.R. Acad. Sci. Paris (Comptes Rendus Mathematique), 339 (2004), 607-610.
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FORMULA
| a(n) = sum{m=0..infinity, binomial(m^2+n-1, n)/2^(m+1)}
G.f.: sum{m=0..infinity, (1-x)^(-m^2)/2^(m+1)}. Row sums of A120945. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 25 2006
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EXAMPLE
| a(1)=3, the three graphs being (1 ->2), (2 ->1) and (1 ->1).
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MAPLE
| d:=proc(n) local m; sum(binomial(m^2+n-1, n)/2^(m+1), m=0..infinity); end;
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MATHEMATICA
| f[n_] := Sum[ Binomial[m^2 + n - 1, n]/2^(m + 1), {m, 0, Infinity}]; Table[ f[n], {n, 0, 15}] (from Robert G. Wilson v Mar 16 2005)
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CROSSREFS
| Cf. A052171 (counts same objects up to labeling).
Cf. A020561.
Sequence in context: A203243 A198970 A014850 * A121247 A064732 A092610
Adjacent sequences: A104206 A104207 A104208 * A104210 A104211 A104212
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KEYWORD
| nonn
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AUTHOR
| Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Mar 13 2005
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EXTENSIONS
| Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 16 2005
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