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 A005333 Number of 2-colored connected labeled graphs with n vertices of the first color and n vertices of the second color. (Formerly M4030) 14
 1, 5, 205, 36317, 23679901, 56294206205, 502757743028605, 17309316971673776957, 2333508400614646874734621, 1243000239291173897659593056765, 2629967962392578020413552363565293565, 22170252073745058975210005804934596601690557 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: if n > 1, then a(n) is the number of labeled digraphs D (allowing self-loops) with n vertices such that D|D' and D'|D are (strongly) connected (see preliminaries of Broere et al.). - Lorenzo Sauras Altuzarra, Sep 17 2022 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Andrew Howroyd, Table of n, a(n) for n = 1..50 I. Broere, W. Imrich, R. Kalinowski, and M. Pilsniak, Asymmetric colorings of products of graphs and digraphs, Discrete Applied Mathematics 266 (p. 56-64), 2019. F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy) FORMULA a(n) = c(n,n) where c(0,1) = 1, c(0,m) = 0, c(n,m) = 2^(n*m) - Sum_{1 <= k <= n, 0 <= j <= m, k < n or j < m} C(n-1, k-1) * C(m, j) * 2^((n-k)*(m-j)) * c(k, j). - Sean A. Irvine, May 11 2016 MATHEMATICA c[0, 1] = c[1, 0] = 1; c[0, _] = c[_, 0] = 0; c[n_, m_] := c[n, m] = 2^(n*m) - Sum[If[k < n || j < m, Binomial[n - 1, k - 1]*Binomial[m, j]* 2^((n - k)*(m - j))*c[k, j], 0], {k, 1, n}, {j, 0, m}]; a[n_] := c[n, n]; Array[a, 12] (* Jean-François Alcover, Sep 03 2019 *) CROSSREFS Main diagonal of A262307 and A227322. Sequence in context: A186722 A060486 A002438 * A162087 A292330 A068531 Adjacent sequences: A005330 A005331 A005332 * A005334 A005335 A005336 KEYWORD nonn AUTHOR N. J. A. Sloane EXTENSIONS More precise definition by Pavel Irzhavski, Jul 09 2013 More terms from Sean A. Irvine, May 11 2016 STATUS approved

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Last modified September 19 09:29 EDT 2024. Contains 376007 sequences. (Running on oeis4.)