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 A000060 Number of signed trees with n nodes. (Formerly M0904 N0340) 4
 1, 2, 3, 10, 27, 98, 350, 1402, 5743, 24742, 108968, 492638, 2266502, 10600510, 50235931, 240882152, 1166732814, 5702046382, 28088787314, 139355139206, 695808554300, 3494391117164, 17641695461662, 89495028762682, 456009893224285, 2332997356507678, 11980753878699716, 61739654456234062, 319188605907760846 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS If only trees with a degree of each node <=2 (linear chains) are counted, we obtain A005418. If only trees with a degree of each node <=3 are counted, we obtain 1, 2, 3, 10, 22, 76, 237, 856... If the degree is restricted to <=4 we obtain 1, 2, 3, 10, 27, 92, 323, 1260,... - R. J. Mathar, Feb 26 2018 REFERENCES 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 T. D. Noe, Table of n, a(n) for n=1..500 F. Harary and G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math., 101 (1959), 141-162. P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy) FORMULA G.f.: S(x)+S(x^2)-S(x)^2, where S(x) is the generating function for A000151. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005 a(n) = A000238(n)+A000151(n/2), where A000151(.) is zero for non-integer arguments. - R. J. Mathar, Apr 16 2018 EXAMPLE For n=4 nodes and 3 edges, the unsigned tree has two forms: the star and the linear chain. The star has 4 ways of signing its 3 edges: without plus (3 minus'), with one plus (2 minus'), with two plusses (1 minus) and with three plusses (no minus).  The linear chain has 6 ways of signing the edges: +++, ---, +-- (equivalent to --+), -++ (equivalent to ++-), -+- and +-+. The total number of ways is a(4) = 4+6=10. - R. J. Mathar, Feb 26 2018 MAPLE unassign('x'): with(combstruct): norootree:=[S, {B = Set(S), S = Prod(Z, B, B)}, unlabeled]: S:=x->add(count(norootree, size=i)*x^i, i=1..30): seq(coeff(S(x)+S(x^2)-S(x)^2, x, i), i=1..29); # with Algolib (Pab Ter) MATHEMATICA b[M_] := Module[{A}, A = Table[1, {M}]; For[n = 1, n <= M-1, n++, A[[n+1]] = 2/n*Sum[Sum[d*A[[d]], {d, Divisors[i]}]*A[[n-i+1]], {i, 1, n}]]; A]; seq[n_] := Module[{g}, g = x*(b[n].x^Range[0, n-1]); CoefficientList[g + (g /. x -> x^2) - g^2, x]][[2 ;; n+1]]; seq (* Jean-François Alcover, Sep 04 2019, after Andrew Howroyd *) PROG (PARI) \\ here b(N) is A000151 as vector b(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); A} seq(n) = {my(g=x*Ser(b(n))); Vec(g + subst(g, x, x^2) - g^2)} \\ Andrew Howroyd, May 13 2018 CROSSREFS Cf. A000151, A000238. Row sums of A302939. Sequence in context: A052929 A151415 A134588 * A089752 A264759 A323680 Adjacent sequences:  A000057 A000058 A000059 * A000061 A000062 A000063 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005 STATUS approved

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Last modified October 4 12:06 EDT 2022. Contains 357239 sequences. (Running on oeis4.)