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A000060 Number of signed trees with n nodes.
(Formerly M0904 N0340)
7
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)

Index entries for sequences related to trees

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)

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 A171190

Adjacent sequences:  A000057 A000058 A000059 * A000061 A000062 A000063

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005

STATUS

approved

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Last modified August 18 06:41 EDT 2018. Contains 313823 sequences. (Running on oeis4.)