1,3

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. - Christian G. Bower, Oct 15 1998

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. - Gary W. Adamson, Dec 04 2008

Equals the eigensequence of A051731, the inverse binomial transform. - Gary W. Adamson, Dec 26 2008

From Emeric Deutsch, Aug 18 2012: (Start)

The considered rooted trees are called generalized Bethe trees; in the Goldberg-Livshitz reference they are called uniform trees.

Also, a(n) = number of partitions of n-1 in which each part is divisible by the next. Example: a(5)=5 because we have 4, 31, 22, 211, and 1111.

There is a simple bijection between generalized Bethe trees with n+1 vertices and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: number of edges --- sum of parts; root degree --- last part; number of leaves --- first part; height --- number of parts. (End)

a(n+1) = a(n) + 1 if and only if n is prime. - Jon Perry, Nov 24 2012

According to the MathOverflow link, log(a(n)) ~ log(4)*log(n)^2, and a more precise asymptotic expansion is similar to that of A018819 and hence A000123, so the conjecture in the Formula section is partly correct. - Andrey Zabolotskiy, Jan 22 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 1..10000

G. Gati, F. Harary, R. W. Robinson, Line colored trees with extendable automorphisms, Acta Mathematica Scientia 2.1 (1982), 105-110. (Annotated scanned copy)

M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)

B. S. Kochkarev, Absolutely symmetric trees and complexity of natural number, arXiv:1205.0344 [math.CO], 2012.

MathOverflow, Are the asymptotics of A003238 known?

O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra and its Appl., 428, 2008, 2961-2979.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Transforms

Gus Wiseman, Planted achiral trees n=1..10.

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Shifts one place left under inverse Moebius transform: a(n+1) = Sum_{k|n} a(k).

Conjecture: log(a(n)) is asymptotic to c*log(n)^2 where 0.4 < c < 0.5 - Benoit Cloitre, Apr 13 2004

For n > 1, a(n) = (1/2) * A068336(n) and Sum_{k = 1..n} a(k) = A003318(n). - Ralf 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

a(n) = 1 + sum of a(i) such that n == 1 (mod i). - Jon Perry, Nov 20 2012

From Ilya Gutkovskiy, Apr 28 2019: (Start)

G.f.: x * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^n)).

L.g.f.: -log(Product_{n>=1} (1 - x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n. (End)

a(4) = 3 because we have the path P(4), the tree Y, and the star \|/ . - Emeric Deutsch, Aug 18 2012

The planted achiral trees with up to 7 nodes are:

1 -

1 (-)

2 (--), ((-))

3 (---), ((--)), (((-)))

5 (----), ((-)(-)), ((---)), (((--))), ((((-))))

6 (-----), ((----)), (((-)(-))), (((---))), ((((--)))), (((((-)))))

10 (------), ((-)(-)(-)), ((--)(--)), (((-))((-))), ((-----)), (((----))), ((((-)(-)))), ((((---)))), (((((--))))), ((((((-)))))). - Gus Wiseman, Jan 12 2017

with(numtheory): aa := proc (n) if n = 0 then 1 else add(aa(divisors(n)[i]-1), i = 1 .. tau(n)) end if end proc: a := proc (n) options operator, arrow: aa(n-1) end proc: seq(a(n), n = 1 .. 48); # Emeric Deutsch, Aug 18 2012

A003238:= proc(n) option remember; uses numtheory; add(A003238(m), m=divisors(n-1)) end proc;

A003238(1):= 1;

[seq(A003238(n), n=1..48)]; # Robert Israel, Mar 10 2014

(* 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}] (* Jean-François Alcover, Dec 20 2011, after Ralf Stephan *)

achi[n_]:=If[n===1, 1, Total[achi/@Divisors[n-1]]]; Array[achi, 50] (* Gus Wiseman, Jan 12 2017 *)

(JavaScript)

a = new Array();

for (i = 1; i < 50; i++) a[i] = 1;

for (i = 3; i < 50; i++) for (j = 2; j < i; j++) if (i % j == 1) a[i] += a[j];

document.write(a + "<br>"); // Jon Perry, Nov 20 2012

(Haskell)

a003238 n = a003238_list !! (n-1)

a003238_list = 1 : f 1 where

f x = (sum (map a003238 $ a027750_row x)) : f (x + 1)

-- Reinhard Zumkeller, Dec 20 2014

Cf. A007439, A007554, A057546, A152434, A051731, A002033, A027750, A281487, A000123.

Row sums of A122934 (offset by 1).

Cf. A004111, A280994.

Sequence in context: A325354 A298363 A018396 * A051839 A130714 A130689

Adjacent sequences: A003235 A003236 A003237 * A003239 A003240 A003241

nonn,nice,eigen

N. J. A. Sloane

Description improved by Christian G. Bower, Oct 15 1998

approved