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 A339830 Number of bicolored trees on n unlabeled nodes such that black nodes are not adjacent to each other. 6
 1, 2, 2, 4, 10, 26, 75, 234, 768, 2647, 9466, 34818, 131149, 503640, 1965552, 7777081, 31138051, 125961762, 514189976, 2115922969, 8769932062, 36584593158, 153510347137, 647564907923, 2744951303121, 11687358605310, 49965976656637, 214423520420723, 923399052307921 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The black nodes form an independent vertex set. For n > 0, a(n) is then the total number of indistinguishable independent vertex sets summed over distinct unlabeled trees with n nodes. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..500 Eric Weisstein's World of Mathematics, Independent Vertex Set EXAMPLE a(2) = 2 because at most one node can be colored black. a(3) = 4 because the only tree is the path graph P_3. If the center node is colored black then neither of the ends can be colored black; otherwise zero, one or both of the ends can be colored black. In total there are 4 possibilities. There are 3 trees with 5 nodes: o o | | o---o---o o---o---o---o---o o---o---o | | o o These correspond respectively to 11, 9 and 6 bicolored trees (with black nodes not adjacent), so a(5) = 11 + 9 + 6 = 26. PROG (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} seq(n)={my(u=v=[1]); for(n=2, n, my(t=concat([1], EulerT(v))); v=concat([1], EulerT(u+v)); u=t); my(g=x*Ser(u+v), gu=x*Ser(u)); Vec(1 + g + (subst(g, x, x^2) - subst(gu, x, x^2) - g^2 + gu^2)/2)} CROSSREFS Cf. A038056 (bicolored trees), A339829, A339831, A339832, A339834, A339837. Sequence in context: A025244 A132824 A298898 * A078801 A309159 A002420 Adjacent sequences: A339827 A339828 A339829 * A339831 A339832 A339833 KEYWORD nonn AUTHOR Andrew Howroyd, Dec 19 2020 STATUS approved

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Last modified September 13 09:59 EDT 2024. Contains 375904 sequences. (Running on oeis4.)