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A224068
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Number of labeled graphs on n vertices that can be colored using exactly 4 colors.
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6
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0, 0, 0, 1536, 122880, 10813440, 1348730880, 261070258176, 81787921367040, 42364317235937280, 36686317873382031360, 53408511909378681470976, 131046345314766385022238720, 542471805171085602081503969280, 3789399960645715708906355231293440
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OFFSET
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1,4
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COMMENTS
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A223887 counts labeled 4-colored graphs on n vertices, that is, colorings of labeled graphs on n vertices using 4 or fewer colors.
This sequence differs in that it counts only those colorings of labeled graphs on n vertices that use exactly 4 colors. Cf. A213441 and A213442.
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LINKS
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FORMULA
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Let E(x) = Sum_{n>=0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + x^4/(4!*2^6) + .... Then a generating function is (E(x) - 1)^4 = 1536*x^4/(4!*2^6) + 122880*x^5/(5!*2^10) + 10813440*x^6/(6!*2^15) + ... + a(n)*x^n/(n!*2^(n*(n-1)/2)) + ... (see Read).
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MATHEMATICA
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nn=20; e[x_]:=Sum[x^n/(n!*2^Binomial[n, 2]), {n, 0, nn}]; Table[n!*2^Binomial[n, 2], {n, 0, nn}]CoefficientList[Series[(e[x]-1)^4, {x, 0, nn}], x] (* Geoffrey Critzer, Aug 11 2014 *)
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PROG
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(PARI)
N=16; x='x+O('x^N);
E=sum(n=0, N, x^n/(n!*2^binomial(n, 2)) );
tgf=(E-1)^4;
v=concat([0, 0, 0], Vec(tgf));
v=vector(#v, n, v[n] * n! * 2^(n*(n-1)/2) )
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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