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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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22
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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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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 = 0.2078615974229174213216534920508516879353537904602582293754027908931077971... - Vaclav Kotesovec, Aug 20 2014, updated Dec 26 2020
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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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terms = 30; A[_] = 0; Do[A[x_] = x*Exp[2*Sum[A[x^k]/k, {k, 1, terms}]] + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest
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PROG
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(PARI) seq(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} \\ Andrew Howroyd, May 13 2018
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CROSSREFS
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KEYWORD
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nonn,eigen,nice
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AUTHOR
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EXTENSIONS
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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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