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A000684
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Number of colored labeled n-node graphs with 2 interchangeable colors.
(Formerly M2954 N1192)
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29
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1, 3, 13, 81, 721, 9153, 165313, 4244481, 154732801, 8005686273, 587435092993, 61116916981761, 9011561121239041, 1882834327457349633, 557257804202631217153, 233610656002563147038721, 138681207656726645785559041
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OFFSET
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1,2
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COMMENTS
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REFERENCES
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R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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
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FORMULA
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G.f.: A(x) = Sum_{n>=1} x^n/(1 - 2^n*x)^n. - Paul D. Hanna, Sep 14 2009
G.f.: 1/(W(0)-x) where W(k) = x*(x*2^k-1)^k - (x*2^(k+1)-1)^(k+1) + x*((2*x*2^k-1)^(2*k+2))/W(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Sep 17 2012
a(n) = Sum_{k = 0..n-1} binomial(n-1,k)*2^(k*(n-k)).
a(n) = Sum_{k = 0..n} 2^k*A111636(n,k).
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence (but with an offset of 0) is E(x)*E(2*x) = Sum_{n >= 0} a(n+1)*x^n/(n!*2^C(n,2)) = 1 + 3*x + 13*x^2/(2!*2) + 81*x^3/(3!*2^3) + 721*x^4/(4!*2^6) + .... Cf. A134531. (End)
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MATHEMATICA
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With[{nn=20}, Rest[CoefficientList[Series[Sum[x^n/(1-2^n x)^n, {n, nn}], {x, 0, nn}], x]]] (* Harvey P. Dale, Nov 24 2011 *)
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PROG
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(PARI) a(n)=polcoeff(sum(k=1, n, x^k/(1-2^k*x +x*O(x^n))^k), n) \\ Paul D. Hanna, Sep 14 2009
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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a(15) onwards added by N. J. A. Sloane, Oct 19 2006 from the Robinson reference
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STATUS
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approved
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