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 A290716 Number of minimal dominating sets in the n-triangular (Johnson) graph. 5
 1, 1, 1, 3, 15, 35, 225, 1197, 6881, 45369, 327375, 2460755, 19925367, 171368067, 1551364997, 14763620445, 147405166785, 1538113071857, 16732908859599, 189413984297187, 2226589748578775, 27130592749003275, 342118450334269917, 4458168165784234253, 59952936723606219009 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A minimal dominating set on the triangular graph corresponds either with a minimal edge cover on the complete graph minus one vertex or with a perfect matching on the complete graph. Perfect matchings on the complete graph exists only for even n. - Andrew Howroyd, Aug 13 2017 Also the number of maximal irredundant sets in the n-triangular graph. - Eric W. Weisstein, Dec 31 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 Eric Weisstein's World of Mathematics, Johnson Graph Eric Weisstein's World of Mathematics, Maximal Irredundant Set Eric Weisstein's World of Mathematics, Minimal Dominating Set Eric Weisstein's World of Mathematics, Triangular Graph FORMULA a(n) = n*A053530(n-1) for n odd, a(n) = (n-1)!! + n*A053530(n-1) for n even. - Andrew Howroyd, Aug 13 2017 E.g.f.: exp(x^2/2) + x*exp(x*exp(x) - (x+x^2/2)). - Andrew Howroyd, Apr 21 2018 MATHEMATICA b[n_]:=n! Sum[1/k! (Binomial[k, n - k] 2^(k - n) (-1)^k + Sum[Binomial[k, j] Sum[j^(i - j)/(i - j)! Binomial[k - j, n - i - k + j] 2^(i - j + k - n) (-1)^(k - j), {i, j, n - k + j}], {j, k}]), {k, n}]; Join[{1, 1}, Table[n b[n - 1] + If[Mod[n, 2] == 0, (n - 1)!!, 0], {n, 2, 20}]] (* Eric W. Weisstein, Aug 14 2017 *) Range[0, 20]! CoefficientList[Series[Exp[x^2/2] + x Exp[x Exp[x] - (x + x^2/2)], {x, 0, 20}], x] (* Eric W. Weisstein, Apr 23 2018 *) PROG (PARI) \\ here b(n) is A053530, df(n) is (2*n-1)!! = A001147 b(n)=polcoeff(serlaplace(exp(-x-1/2*x^2+x*exp(x+O(x^(n+1))))), n, x); df(n)=polcoeff(serlaplace((1-2*x+O(x^(n+1)))^(-1/2)), n, x); a(n) = n*b(n-1) + if(n%2==0, df(n/2), 0); \\ Andrew Howroyd, Aug 13 2017 (PARI) seq(n)={Vec(serlaplace(exp(x^2/2 + O(x*x^n)) + x*exp(x*exp(x + O(x^n)) - (x+x^2/2))))} \\ Andrew Howroyd, Apr 21 2018 CROSSREFS Cf. A001147, A053530, A290847. Sequence in context: A187787 A290717 A019009 * A347998 A162441 A001803 Adjacent sequences: A290713 A290714 A290715 * A290717 A290718 A290719 KEYWORD nonn AUTHOR Eric W. Weisstein, Aug 09 2017 EXTENSIONS a(8)-a(24) from formula by Andrew Howroyd, Aug 13 2017 a(0)-a(1) prepended by Andrew Howroyd, Apr 21 2018 STATUS approved

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Last modified March 23 03:40 EDT 2023. Contains 361434 sequences. (Running on oeis4.)