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 A000686 Number of 4-colored labeled graphs on n nodes, divided by 4. (Formerly M4449 N1884) 4
 1, 7, 85, 1777, 63601, 3882817, 403308865, 71139019777, 21276992674561, 10778161937857537, 9238819435213784065, 13390649605615389843457, 32796747486424209782108161, 135669064080920007649863745537, 947468281528010179181982467702785, 11166618111585805201637975219611631617 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Sequence represents 1/4 of the number of 4-colored labeled graphs on n nodes. Indeed, on p. 413 of the Read paper, column 4 is 4, 28, 340, 7108, ... - Emeric Deutsch, May 06 2004 REFERENCES R. C. Read, personal communication. 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). LINKS S. R. Finch, Bipartite, k-colorable and k-colored graphs (4*A000686) R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410—414. R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976 FORMULA a(n) = (1/4)Sum_{k=0..n} binomial(n, k)*2^(k(n-k))*b(k), where b(0)=1 and b(k)=3*A000685(k) for k > 0. - Emeric Deutsch, May 06 2004 From Peter Bala, Apr 12 2013: (Start) a(n) = (1/4)*A223887(n). a(n) = (1/4)*Sum_{k = 0..n} binomial(n,k)*2^(k*(n-k))*b(k)*b(n-k), where b(n) := Sum_{k = 0..n} binomial(n,k)*2^(k*(n-k)). Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is 1/4*(E(x)^4 - 1) = Sum_{n >= 0} a(n)*x^n/(n!*2^C(n,2)) = x + 7*x^2/(2!*2) + 85*x^3/(3!*2^3) + .... (End) MATHEMATICA b[n_] := Sum[ 2^((i-j)*j + i*(n-i))*Binomial[n, i]*Binomial[i, j], {i, 0, n}, {j, 0, i}]; a[n_] := 1/4*Sum[ Binomial[n, k]*2^(k*(n-k))*b[k], {k, 0, n}]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 07 2011, after Emeric Deutsch *) PROG (PARI) N=66;  x='x+O('x^N); E=sum(n=0, N, x^n/(n!*2^binomial(n, 2)) ); tgf=E^4-1;  v=Vec(tgf); v=vector(#v, n, v[n] * n! * 2^(n*(n-1)/2) ) / 4 /* Joerg Arndt, Apr 10 2013 */ CROSSREFS Cf. A000683, A000685, A223887. Sequence in context: A060237 A000424 A207214 * A102923 A220246 A196257 Adjacent sequences:  A000683 A000684 A000685 * A000687 A000688 A000689 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Pab Ter (pabrlos(AT)yahoo.com) and Emeric Deutsch, May 05 2004 STATUS approved

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Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)