|
|
A003238
|
|
Number of rooted trees with n vertices in which vertices at the same level have the same degree.
(Formerly M0628)
|
|
195
|
|
|
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
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
|
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).
|
|
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
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
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;
|
|
MATHEMATICA
|
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 + "<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)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice,eigen
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|