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 A003238 Number of rooted trees with n vertices in which vertices at the same level have the same degree. (Formerly M0628) 189
 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; text; internal format)
 OFFSET 1,3 COMMENTS 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS 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. FORMULA 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) EXAMPLE 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 MAPLE 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 MATHEMATICA (* b = A068336 *) b = 1; b[n_] := b[n] = 1 + Sum[b[k], {k, Divisors[n-1]}]; a[n_] := b[n]/2; a = 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 *) PROG (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 + "
"); // 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 CROSSREFS 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 KEYWORD nonn,nice,eigen AUTHOR EXTENSIONS Description improved by Christian G. Bower, Oct 15 1998 STATUS approved

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Last modified September 27 09:05 EDT 2022. Contains 357054 sequences. (Running on oeis4.)