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A000151
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Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.
(Formerly M1770 N0701)
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11
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1, 2, 7, 26, 107, 458, 2058, 9498, 44947, 216598, 1059952, 5251806, 26297238, 132856766, 676398395, 3466799104, 17873508798, 92630098886, 482292684506, 2521610175006, 13233573019372, 69687684810980, 368114512431638, 1950037285256658, 10357028326495097, 55140508518522726, 294219119815868952, 1573132563600386854, 8427354035116949486, 45226421721391554194
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OFFSET
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1,2
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 307 and 564.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, R(x).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
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).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..500
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 387
P. Leroux and B. Miloudi, Generalisations de la formule d'Otter, Ann. Sci. Math. Quebec 16 (1992), no 1, 53-80.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077, 2016.
S. G. Wagner, An identity for the cycle indices of rooted tree automorphism groups, Elec. J. Combinat., 13 (2006), #R00.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
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FORMULA
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Generating function A(x) = x+2*x^2+7*x^3+26*x^4+... satisfies A(x)=x*exp( 2*sum_{k>=1}(A(x^k)/k) ) [Harary]. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005
G.f.: x*Product_{n>=1} 1/(1 - x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n.
a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.64654261623294971289271351621..., c = 0.207861597422917421321653492... . - Vaclav Kotesovec, Aug 20 2014
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MAPLE
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R:=series(x+2*x^2+7*x^3+26*x^4, x, 5); M:=500;
for n from 5 to M do
series(add( subs(x=x^k, R)/k, k=1..n-1), x, n);
t4:=coeff(series(x*exp(%)^2, x, n+1), x, n);
R:=series(R+t4*x^n, x, n+1); od:
for n from 1 to M do lprint(n, coeff(R, x, n)); od: # N. J. A. Sloane, Mar 10 2007
with(combstruct):norootree:=[S, {B = Set(S), S = Prod(Z, B, B)}, unlabeled] :seq(count(norootree, size=i), i=1..30); # with Algolib (Pab Ter)
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MATHEMATICA
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n = 30; d[x_] = Sum[c[k] x^k, {k, 2, 2 n, 2}];
se = Sum[Series[Sum[(2/m)*d[x^m], {m, 1, n}], {x, 0, 2*n}]^k/k!, {k, 0, n}] - Series[d[x]/x^2, {x, 0, 2*n}];
eq = Thread[CoefficientList[se, x] == 0] // Union // Rest;
Do[so[k] = Solve[eq[[k]]] // First; eq = eq /. so[k], {k, 1, n}]
Table[c[k] , {k, 2, 2 n, 2}] /. Flatten[Array[so, n]]
(* Jean-François Alcover, Jun 08 2011, after g.f. *)
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CROSSREFS
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Cf. A000238, A038055.
Also the self-convolution of A005750. - Paul D. Hanna, Aug 17 2002
Column k=2 of A242249.
Cf. A005751, A245870.
Sequence in context: A150565 A150566 A150567 * A150568 A102319 A006603
Adjacent sequences: A000148 A000149 A000150 * A000152 A000153 A000154
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KEYWORD
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nonn,eigen,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Extended with alternate description by Christian G. Bower, Apr 15 1998
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005
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STATUS
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approved
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