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A003238
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Number of rooted trees where at each node all sub-rooted trees are identical.
(Formerly M0628)
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18
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1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 27, 40, 41, 53, 61, 77, 78, 104, 105, 134, 147, 175, 176, 227, 233, 275, 294, 350, 351, 438, 439, 516, 545, 624, 640, 774, 775, 881, 924, 1069, 1070, 1265, 1266, 1444, 1521, 1698, 1699
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Comment from Christian Bower: also, number of sequences of positive integers a_1,a_2,...,a_k such that 1+a_1*(1+a_2*(...(1+a_k)...))=n. If you take mu(a_1)*mu(a_2)*...*mu(a_k) for each sequence you get 1's 0's and -1's. Add them up and you get the terms for A007554.
Note that this applies also to planar rooted trees and other similar objects (mountain ranges, parenthesizations) encoded by A014486 - Antti Karttunen Sep 07 2000
Equals sum of (n-1)-th row terms of triangle A152434 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 04 2008]
Equals the eigensequence of A051731, the inverse binomial transform. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 26 2008]
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REFERENCES
| G. Gati, F. Harary and R. W. Robinson, Line colored trees with extendable automorphisms, Acta Math. Scientia, 2 (1982), 105-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
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FORMULA
| Shifts one place left under inverse Moebius transform: a(n+1)= Sum a(k), k|n.
Conjecture : log(a(n)) is asymptotic to c*log(n)^2 where 0.4<c<0.5 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 13 2004
For n>1, a(n) = 1/2 * A068336(n) and sum(k=1, n, a(k)) = A003318(n). - R. Stephan, Mar 27 2004
Generating function P(x) for the sequence with offset 2 obeys P(x) = x^2*(1+sum_{n>=1} P(x^n)/x^n). [Harary & Robinson]. - R. J. Mathar, Sep 28 2011
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MATHEMATICA
| (* b = A068336 *) b[1] = 1; b[n_] := b[n] = 1 + Sum[b[k], {k, Divisors[n-1]}]; a[n_] := b[n]/2; a[1] = 1; Table[ a[n], {n, 1, 48}] (* From Jean-François Alcover, Dec 20 2011, after R. Stephan *)
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CROSSREFS
| Cf. A007439, A007554, A057546.
Row sums of A122934 (offset by 1).
A152434 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 04 2008]
A051731 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 26 2008]
Sequence in context: A074243 A072720 A018396 * A051839 A130714 A130689
Adjacent sequences: A003235 A003236 A003237 * A003239 A003240 A003241
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KEYWORD
| nonn,nice,eigen
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Description improved by Christian G. Bower (bowerc(AT)usa.net), Oct 15 1998.
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