A068336 -id:A068336 - cp's OEIS frontend

A003238 Number of rooted trees with n vertices in which vertices at the same level have the same degree.

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

Offset: 1

N. J. A. Sloane

Also, number of sequences of positive integers b_1, b_2, ..., b_k such that 1 + b_1*(1 + b_2*(...(1 + b_k) ... )) = n. If you take mu(b_1)*mu(b_2)*...*mu(b_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

			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

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, and 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

Cf. A007439, A007554, A057546, A152434, A051731, A002033, A027750, A281487, A000123.
Row sums of A122934 (offset by 1).
Cf. A004111, A280994.

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

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

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 *)

seq(n) = {my(v=vector(n)); v[1]=1; for(i=2, n, v[i]=sumdiv(i-1, d, v[d])); v} \\ Andrew Howroyd, Jun 08 2025

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)

Description improved by Christian G. Bower, Oct 15 1998

A339755 a(1) = 1; a(n+1) = 1 + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 2, 5, 11, 27, 55, 131, 263, 571, 1168, 2445, 4891, 10113, 20227, 40979, 82229, 165632, 331265, 665365, 1330731, 2666729, 5334769, 10679319, 21358639, 42740683, 85482096, 171004645, 342015001, 684113793, 1368227587, 2736633741, 5473267483, 10946869669, 21893763789, 43788190107

Offset: 1

Ilya Gutkovskiy, Dec 15 2020

nonn

Cf. A038044, A068336, A097558, A122698, A277120, A325303.

a:= proc(n) option remember; uses numtheory;
      1+add(a(d)*a((n-1)/d), d=divisors(n-1))
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Dec 15 2020

a[1] = 1; a[n_] := a[n] = 1 + Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 35}]

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

a(n) ~ c * 2^n, where c = 1.27442410710035207761153205319824525254716841098942446508584158048310907298... - Vaclav Kotesovec, Dec 16 2020

A343371 a(n) = 1 + Sum_{d|n, d < n} a(d - 1).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 9, 2, 8, 7, 7, 2, 12, 2, 12, 9, 9, 2, 13, 5, 12, 9, 12, 2, 22, 2, 14, 10, 10, 10, 18, 2, 15, 13, 16, 2, 26, 2, 20, 20, 12, 2, 22, 7, 23, 11, 19, 2, 26, 11, 23, 16, 15, 2, 30, 2, 25, 26, 16, 14, 36, 2, 22, 13, 27, 2, 32, 2, 21, 28

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A003238, A067824, A068336, A167865.

a:= proc(n) option remember;
      1+add(a(d-1), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=0..75);  # Alois P. Heinz, Apr 12 2021

a[n_] := a[n] = 1 + Sum[If[d < n, a[d - 1], 0], {d, Divisors[n]}]; Table[a[n], {n, 0, 75}]
nmax = 75; A[] = 0; Do[A[x] = 1/(1 - x) + Sum[x^k A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^2 * A(x^2) + x^3 * A(x^3) + x^4 * A(x^4) + ...

A068334 a(1) = 1; a(n+1) = 1 + Product_{k|n} a(k), product is over the positive divisors, k, of n.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 61, 62, 497, 1492, 26857, 26858, 6445921, 6445922, 786402485, 21232867096, 10531502079617, 10531502079618, 314049392014208761, 314049392014208762, 33736441887734362049089, 6173768865455388254983288

Offset: 1

Leroy Quet, Feb 27 2002

nonn

			a(7) = 1 + a(1)*a(2)*a(3)*a(6) = 1 + 1*2*3*10 = 61.

Seiichi Manyama, Table of n, a(n) for n = 1..66

Cf. A068336, A343390.

Nest[Append[#1, 1 + Times @@ #1[[Divisors[#2] ]] ] & @@ {#, Length[#]} &, {1}, 21] (* Michael De Vlieger, Apr 13 2021 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(d=divisors(n-1)); va[n] = 1 + prod(k=1, #d, va[d[k]]);); va;} \\ Michel Marcus, Apr 13 2021

A160183 Triangle, read by rows, obtained by dropping right diagonal from A160182.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 6, 2, 1, 1, 1, 10, 4, 2, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 16, 6, 3, 2, 1, 1, 1, 1, 19, 7, 4, 2, 1, 1, 1, 1, 1, 26, 10, 5, 3, 2, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, May 03 2009

A054525 * A160183 = triangle A160182. Mobius transform A054525 of the triangle shifts to the left adding I, I = Identity matrix.

Left border = A003238 starting (1, 2, 3, 5, 6, 10, 11, 16, 19,...).

			First few rows of the triangle =
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
6, 2, 1, 1, 1;
10, 4, 2, 1, 1, 1;
11, 4, 2, 1, 1, 1, 1;
16, 6, 3, 2, 1, 1, 1, 1;
19, 7, 4, 2, 1, 1, 1, 1, 1;
26, 10, 5, 3, 2, 1, 1, 1, 1, 1;
...

Cf. A160182, A068336.

A054525 * triangle A160182, also obtained by shifting triangle A160182 to the right, deleting the (1,1,1,...) right border.

A160182 Triangle read by rows, 1 / ((-1)A129184 A051731 + I), I = Identity matrix.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 6, 2, 1, 1, 1, 1, 10, 4, 2, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 1, 16, 6, 3, 2, 1, 1, 1, 1, 1, 19, 7, 4, 2, 1, 1, 1, 1, 1, 1, 26, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, May 03 2009

Inverse mobius transform (A051731) * the triangle shifts row terms to the right deleting the right border, getting triangle A160183: (1; 2,1; 3,1,1; 5,2,1,1;...).

			First few rows of the triangle:
   1;
   1,  1;
   2,  1,  1;
   3,  1,  1,  1;
   5,  2,  1,  1,  1;
   6,  2,  1,  1,  1,  1;
  10,  4,  2,  1,  1,  1,  1;
  11,  4,  2,  1,  1,  1,  1,  1;
  16,  6,  3,  2,  1,  1,  1,  1,  1;
  19,  7,  4,  2,  1,  1,  1,  1,  1,  1;
  26, 10,  5,  3,  2,  1,  1,  1,  1,  1,  1;
  ...

Cf. A051731, A129184, A160183.

Row sums = A068336. Left border = A003238.

A160182den := proc(n,k)
    a := add( A129184(n,i)*A051731(i,k),i=1..n) ;
    if n =k then
        -a+1 ;
    else
        -a;
    end if;
end proc:
N := 20 :
M := Matrix(N,N) :
for n from 1 to N do
for k from 1 to N do
    M[n,k] := A160182den(n,k) ;
end do:
end do:
MatrixInverse(M) ;  # R. J. Mathar, Aug 04 2015

Triangle read by rows, 1 / ((-1)*A129184 * A051731 + I), I = Identity matrix. The operations shift the inverse Mobius transform (A051731) down, changing the signs to (-1), then add I = (1,1,1,...) as the right border.

A290845 a(1) = 1; a(n) = Sum_{k=1..n} a(ceiling((n-1)/k)).

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 36, 56, 78, 110, 148, 200, 254, 334, 416, 522, 644, 798, 954, 1162, 1372, 1640, 1934, 2284, 2636, 3090, 3556, 4106, 4694, 5394, 6096, 6972, 7850, 8882, 9972, 11220, 12500, 14048, 15598, 17360, 19208, 21346, 23486, 26016, 28548, 31436, 34478, 37874, 41272, 45246

Offset: 1

Ilya Gutkovskiy, Aug 12 2017

nonn

			a(1) = 1;
a(2) = a(ceiling(1/1)) + a(ceiling(1/2)) = a(1) + a(1) = 2;
a(3) = a(ceiling(2/1)) + a(ceiling(2/2)) + a(ceiling(2/3)) = a(2) + a(1) + a(1) = 4, etc.

Cf. A003318, A025523, A068336 (first differences), A078346.

a[1] = 1; a[n_] := a[n] = Sum[a[Ceiling[(n - 1)/k]], {k, 1, n}]; Table[a[n], {n, 50}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A290845(n):
    if n == 1:
        return 1
    c, j, k1 = n, 1, n-2
    while k1 > 1:
        j2 = (n-2)//k1 + 1
        c += (j2-j)*A290845(k1+1)>>1
        j, k1 = j2, (n-2)//j2
    return c-j<<1 # Chai Wah Wu, Apr 29 2025

a(n) = 2*A003318(n-1) for n > 1.

A346117 a(1) = a(2) = 1; a(n+2) = 1 + Sum_{d|n} a(d).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 11, 8, 17, 12, 24, 14, 38, 16, 47, 24, 64, 26, 83, 28, 110, 38, 125, 40, 174, 46, 191, 58, 241, 60, 289, 62, 353, 78, 380, 90, 490, 92, 519, 110, 640, 112, 723, 114, 851, 146, 892, 148, 1113, 156, 1177, 184, 1371, 186, 1500, 204, 1752, 234, 1813

Offset: 1

Ilya Gutkovskiy, Jul 05 2021

nonn

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

Cf. A007439, A068336.

f:= proc(n) option remember; local d; 1 + add(procname(d), d = numtheory:-divisors(n-2)) end proc:
f(1):= 1: f(2):= 1:
map(f, [$1..60]); # Robert Israel, Dec 02 2022

a[1] = a[2] = 1; a[n_] := a[n] = 1 + Sum[a[d], {d, Divisors[n - 2]}]; Table[a[n], {n, 1, 60}]
nmax = 60; A[] = 0; Do[A[x] = x + x^2 (1/(1 - x) + Sum[A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x + x^2 * (1 / (1 - x) + A(x) + A(x^2) + A(x^3) + ...).

cp's OEIS Frontend

A226175 a(n) = A068336(n+1) - 1.

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A003238 Number of rooted trees with n vertices in which vertices at the same level have the same degree.

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A339755 a(1) = 1; a(n+1) = 1 + Sum_{d|n} a(n/d) * a(d).

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A343371 a(n) = 1 + Sum_{d|n, d < n} a(d - 1).

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A068334 a(1) = 1; a(n+1) = 1 + Product_{k|n} a(k), product is over the positive divisors, k, of n.

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A160183 Triangle, read by rows, obtained by dropping right diagonal from A160182.

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A160182 Triangle read by rows, 1 / ((-1)*A129184 * A051731 + I), I = Identity matrix.

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A290845 a(1) = 1; a(n) = Sum_{k=1..n} a(ceiling((n-1)/k)).

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A160182 Triangle read by rows, 1 / ((-1)A129184 A051731 + I), I = Identity matrix.