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 A047863 Number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa. 38
 1, 2, 6, 26, 162, 1442, 18306, 330626, 8488962, 309465602, 16011372546, 1174870185986, 122233833963522, 18023122242478082, 3765668654914699266, 1114515608405262434306, 467221312005126294077442, 277362415313453291571118082 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums of A111636. - Peter Bala, Sep 30 2012 Column 2 of Table 2 in Read. - Peter Bala, Apr 11 2013 It appears that 5 does not divide a(n), that a(n) is even for n>0, that 3 divides a(2n) for n>0, that 7 divides a(6n+5), and that 13 divides a(12n+3). - Ralf Stephan, May 18 2013 REFERENCES H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 79, Eq. 3.11.2. LINKS T. D. Noe, Table of n, a(n) for n = 0..50 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. S. R. Finch, Bipartite, k-colorable and k-colored graphs S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author] A. Gainer-Dewar and I. M. Gessel, Enumeration of bipartite graphs and bipartite blocks, arXiv:1304.0139 [math.CO], 2013. D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy] Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414. R. P. Stanley, Acyclic orientation of graphs Discrete Math. 5 (1973), 171-178. North Holland Publishing Company. Eric Weisstein's World of Mathematics, k-Colorable Graph H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2. FORMULA a(n) = Sum_{k=0..n} binomial(n, k)*2^(k*(n-k)). a(n) = 4 * A000683(n) + 2. - Vladeta Jovovic, Feb 02 2000 E.g.f.: Sum_{n>=0} exp(2^n*x)*x^n/n!. - Paul D. Hanna, Nov 27 2007 O.g.f.: Sum_{n>=0} x^n/(1 - 2^n*x)^(n+1). - Paul D. Hanna, Mar 08 2008 From Peter Bala, Apr 11 2013: (Start) Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function is E(x)^2 = 1 + 2*x + 6*x^2/(2!*2) + 26*x^3/(3!*2^3) + .... In general, E(x)^k, k = 1, 2, ..., is a generating function for labeled k-colored graphs (see Stanley). For other examples see A191371 (k = 3) and A223887 (k = 4). If A(x) = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + ... denotes the e.g.f. for this sequence then sqrt(A(x)) = 1 + x + 2*x^2/2! + 7*x^3/3! + ... is the e.g.f. for A047864, which counts labeled 2-colorable graphs. (End) a(n) ~ c * 2^(n^2/4+n+1/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 2^(-k^2) = EllipticTheta[3, 0, 1/2] = 2.128936827211877... if n is even and c = Sum_{k = -infinity..infinity} 2^(-(k+1/2)^2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302... if n is odd. - Vaclav Kotesovec, Jun 24 2013 EXAMPLE For n=2, {1,2 black, not connected}, {1,2 white, not connected}, {1 black, 2 white, not connected}, {1 black, 2 white, connected}, {1 white, 2 black, not connected}, {1 white, 2 black, connected}. G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 162*x^4 + 1442*x^5 + 18306*x^6 + ... MATHEMATICA Table[Sum[Binomial[n, k]2^(k(n-k)), {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, May 09 2012 *) nmax = 20; CoefficientList[Series[Sum[E^(2^k*x)*x^k/k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 05 2019 *) PROG (PARI) {a(n)=n!*polcoeff(sum(k=0, n, exp(2^k*x +x*O(x^n))*x^k/k!), n)} \\ Paul D. Hanna, Nov 27 2007 (PARI) {a(n)=polcoeff(sum(k=0, n, x^k/(1-2^k*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Mar 08 2008 (PARI) N=66; x='x+O('x^N); egf = sum(n=0, N, exp(2^n*x)*x^n/n!); Vec(serlaplace(egf))  \\ Joerg Arndt, May 04 2013 (Python) from sympy import binomial def a(n): return sum([binomial(n, k)*2**(k*(n - k)) for k in range(n + 1)]) # Indranil Ghosh, Jun 03 2017 CROSSREFS Column k=2 of A322280. Cf. A001831, A002031, A052332. Cf. A135079 (variant). Cf. A111636. A033995, A047864, A191371, A223887. Sequence in context: A135922 A213430 A103367 * A180349 A141713 A005272 Adjacent sequences:  A047860 A047861 A047862 * A047864 A047865 A047866 KEYWORD nonn,nice AUTHOR EXTENSIONS Better description from Christian G. Bower, Dec 15 1999 STATUS approved

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Last modified November 23 14:26 EST 2020. Contains 338590 sequences. (Running on oeis4.)