A007554 -id:A007554 - 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

A038046 Shifts left under transform T where Ta is (identity) DCONV a.

Original entry on oeis.org

1, 1, 3, 6, 12, 17, 32, 39, 63, 81, 120, 131, 213, 226, 311, 377, 503, 520, 742, 761, 1031, 1169, 1442, 1465, 2008, 2093, 2558, 2801, 3465, 3494, 4591, 4622, 5628, 6054, 7111, 7390, 9321, 9358, 10899, 11616, 13873, 13914, 17070, 17113, 20063, 21509, 24462

Offset: 1

Christian G. Bower

Eigensequence of triangle A126988. (i.e. the sequence shifts upon multiplication from the left by triangle A126988). - Gary W. Adamson, Apr 27 2009

Number of planted achiral trees with a distinguished leaf. - Gus Wiseman, Jul 31 2018

			From _Gus Wiseman_, Jul 31 2018: (Start)
The a(5) = 12 planted achiral trees with a distinguished leaf:
  (Oooo), (oOoo), (ooOo), (oooO),
  ((O)(o)), ((o)(O)),
  ((Ooo)), ((oOo)), ((ooO)),
  (((Oo))), (((oO))),
  ((((O)))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A126988. - Gary W. Adamson, Apr 27 2009

Cf. A000081, A002033, A003238, A004111, A007554, A038046, A067824, A317580.

a:= proc(n) option remember; `if`(n<2, n, (m-> m*
      add(a(d)/d, d=numtheory[divisors](m)))(n-1))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 09 2019

a[n_]:=If[n==1,1,Sum[d*a[(n-1)/d],{d,Divisors[n-1]}]];
Array[a,30] (* Gus Wiseman, Jul 31 2018 *)

a(1) = 1; a(n > 1) = Sum_{d|(n-1)} d * a((n-1)/d). - Gus Wiseman, Jul 31 2018

G.f. A(x) satisfies: A(x) = x * (1 + Sum_{j>=1} j*A(x^j)). - Ilya Gutkovskiy, May 09 2019

A007557 Shifts left when inverse Moebius transform applied twice.

Original entry on oeis.org

1, 1, 3, 5, 10, 12, 24, 26, 43, 52, 78, 80, 133, 135, 189, 219, 295, 297, 428, 430, 584, 642, 804, 806, 1100, 1123, 1395, 1494, 1856, 1858, 2428, 2430, 2977, 3143, 3739, 3811, 4790, 4792, 5654, 5930, 7072, 7074, 8656

Offset: 1

N. J. A. Sloane

Equals eigensequence of triangle A127170 (the square of the inverse Mobius transform). - Gary W. Adamson, Apr 27 2009

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A000005, A007554, A038045, A038046, A003238, A070965.

Cf. A127170. - Gary W. Adamson, Apr 27 2009

a[n_] := a[n] = Sum[ DivisorSigma[0, (n - 1)/d]*a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 43}] (* Jean-François Alcover, Dec 12 2011, after Vladeta Jovovic *)

a(n+1) = Sum_{d divides n} tau(n/d)*a(d). - Vladeta Jovovic, Jan 24 2003
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*x^(i*j)/(1 - x^(i*j))). (End)

More terms from Vladeta Jovovic, Jan 24 2003

A318583 a(1) = a(2) = 1; for n > 2, a(n+2) = Sum_{d|n} mu(n/d)*a(d).

Original entry on oeis.org

1, 1, 1, 0, 0, -1, -1, -2, -2, -2, -3, -2, -4, 0, -5, 1, -5, 3, -6, 7, -7, 10, -6, 13, -7, 17, -7, 21, -5, 22, -6, 31, -7, 30, -4, 35, -2, 33, -3, 39, 1, 34, 0, 42, -1, 33, 7, 39, 6, 23, 7, 32, 12, 16, 11, 18, 15, -1, 21, 4, 20, -27, 19, -21, 29, -52, 34, -56, 33, -85, 39, -80, 38, -130, 37

Offset: 1

Ilya Gutkovskiy, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
N. J. A. Sloane, Transforms

Cf. A007439, A007554, A008683.

with(numtheory): P:=proc(q) local k,n,x; x:=[1,1]: for n from 3 to q do
x:=[op(x),add(mobius((n-2)/k)*x[k],k=divisors(n-2))]; od; op(x); end:
P(75); # Paolo P. Lava, May 15 2019

a[1] = a[2] = 1; a[n_] := a[n] = Sum[a[d] MoebiusMu[(n - 2)/d], {d, Divisors[n - 2]}]; Table[a[n], {n, 75}]

A318583(n) = if(n<=2,1,sumdiv(n-2,d,moebius((n-2)/d)*A318583(d))); \\ (A non-memoized implementation) - Antti Karttunen, Aug 29 2018

\\ A faster implementation:
up_to = 16384;
A318583list(up_to) = { my(u=vector(up_to)); u[1] = u[2] = 1; for(n=3, up_to, u[n] = sumdiv(n-2,d,moebius((n-2)/d)*u[d])); (u); };
v318583 = A318583list(up_to);
A318583(n) = v318583[n]; \\ Antti Karttunen, Aug 29 2018

G.f.: Sum_{n>=1} a(n+2)*x^n/(1 - x^n).

L.g.f.: -log(Product_{n>=1} (1 - x^n)^(a(n+2)/n)) = Sum_{n>=1} a(n)*x^n/n.

A317581 a(1) = 1; a(n > 1) = 1 + Sum_{d|n, d

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, -2, 0, 2, 2, 1, 0, -2, 0, -2, 2, 2, 0, 4, 1, 2, 0, -2, 0, -6, 0, 0, 2, 2, 2, 7, 0, 2, 2, 4, 0, -6, 0, -2, -2, 2, 0, -4, 1, -2, 2, -2, 0, 4, 2, 4, 2, 2, 0, 16, 0, 2, -2, 1, 2, -6, 0, -2, 2, -6, 0, -12, 0, 2, -2, -2, 2, -6, 0, -4

Offset: 1

Gus Wiseman, Jul 31 2018

If p is prime, a(p^k) = 0 if k is odd, 1 if k is even. - Robert Israel, Aug 01 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001678, A002033, A007554, A038046, A067824.

f:= n -> 1 + add(numtheory:-mobius(n/d)*procname(d),d=numtheory:-divisors(n) minus {n}):
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Aug 01 2018

a[n_]:=1+Sum[MoebiusMu[n/d]*a[d],{d,Most[Divisors[n]]}];
Array[a,100]

from sympy import mobius, divisors
def A317581(n): return 1 + (0 if n == 1 else sum(mobius(n//d)*A317581(d) for d in divisors(n,generator=True) if d < n)) # Chai Wah Wu, Jan 14 2022

A343188 a(1) = a(2) = a(3) = 1; a(n+3) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 0, -1, -1, -1, -2, -2, -1, -3, -2, -2, -3, -2, -1, -4, 0, -2, -3, 0, 0, -4, 4, 0, -3, 5, 3, -4, 9, 2, -2, 11, 5, -1, 15, 4, 0, 16, 10, -1, 20, 9, 1, 24, 12, 0, 25, 12, 1, 28, 16, 0, 25, 19, 2, 26, 22, 1, 26, 21, -2, 28, 25, 0, 20, 24, -2, 23, 30, -3, 10

Offset: 1

Ilya Gutkovskiy, Apr 07 2021

sign

Cf. A007554, A307993, A318583, A343189, A343190.

a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[MoebiusMu[(n - 3)/d] a[d], {d, Divisors[n - 3]}]; Table[a[n], {n, 75}]

A343189 a(1) = ... = a(4) = 1; a(n+4) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, -2, -2, -1, -2, -2, -3, -1, -3, -1, -4, 0, -3, 0, -5, 2, -4, 1, -5, 4, -5, 6, -6, 5, -4, 8, -6, 8, -5, 11, -5, 10, -6, 17, -6, 11, -3, 21, -7, 12, -3, 26, -5, 12, -4, 33, -4, 11, -1, 37, -5, 11, -2, 42, -1, 7, -1, 48, -2, 1, 3, 58, -3

Offset: 1

Ilya Gutkovskiy, Apr 07 2021

sign

Cf. A007554, A307994, A318583, A343188, A343190.

a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = Sum[MoebiusMu[(n - 4)/d] a[d], {d, Divisors[n - 4]}]; Table[a[n], {n, 75}]

A343190 a(1) = ... = a(5) = 1; a(n+5) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -2, -2, -1, -2, -1, -3, -2, -2, -2, -1, -3, -3, 0, -3, 0, -3, -2, -1, -1, -1, -2, -2, 1, -1, 3, -3, -1, 2, 2, 2, -1, -2, 5, 4, 4, -2, 1, 5, 7, 6, -1, 0, 10, 7, 10, 0, 0, 9, 14, 9, 0, 2, 12, 15, 14, -1, 3, 14, 18

Offset: 1

Ilya Gutkovskiy, Apr 07 2021

sign

Cf. A007554, A307995, A318583, A343188, A343189.

a[1] = a[2] = a[3] = a[4] = a[5] = 1; a[n_] := a[n] = Sum[MoebiusMu[(n - 5)/d] a[d], {d, Divisors[n - 5]}]; Table[a[n], {n, 75}]

A143809 Eigentriangle of the Mobius transform, (A054525).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, -3, -1, 0, 0, 0, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, -3, 1, -1, 0, 0, 2, 0, 0, 0, 0, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 3, 0, 0, 0, 0, 0, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -1, 0, 0, 0

Offset: 1

Gary W. Adamson, Sep 01 2008

The eigentriangle of the Mobius transform may be defined by the operation consisting of the termwise product of A054525 row terms and the first n terms of A007554, where A007554: (1, 1, 0, -1, -2, -3, -3,...) = the eigensequence of A054525.
This triangle has the following properties:
Sum of n-th row terms = rightmost term of next row.
Right border = A007554, the eigensequence of the Mobius transform.
Row sums = A007554 shifted one place to the left: (1, 0, -1, -2, -3,...).
Left border = mu(n), A008683.
A054525 = the Mobius transform and A007554 = the eigensequence of A054525.

			First few rows of the triangle:
   1;
  -1,  1;
  -1,  0, 0;
   0, -1, 0, -1;
  -1,  0, 0,  0, -2;
   1, -1, 0,  0,  0, -3;
  -1,  0, 0,  0,  0,  0, -3;
   0,  0, 0,  1,  0,  0,  0, -4;
   0,  0, 0,  0,  0,  0,  0,  0, -3;
   1, -1, 0,  0,  2,  0,  0,  0,  0, -3;
   ...
Row 6 = (1, -1, 0, 0, 0, -3) = termwise product of row 6 of the Mobius transform (1, -1, -1, 0, 0, 1) and the first 6 terms of A007554, (the eigensequence of the Mobius transform): (1, 1, 0, -1, -2, -3).

Cf. A008683, A007554, A054525.

Triangle read by rows, A054525 * (A007554 * 0^(n-k)); 1<=k<=n

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

JavaScript

Maple

Mathematica

PARI

Formula

Extensions

A038046 Shifts left under transform T where Ta is (identity) DCONV a.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A007557 Shifts left when inverse Moebius transform applied twice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A318583 a(1) = a(2) = 1; for n > 2, a(n+2) = Sum_{d|n} mu(n/d)*a(d).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A317581 a(1) = 1; a(n > 1) = 1 + Sum_{d|n, d

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

A343188 a(1) = a(2) = a(3) = 1; a(n+3) = Sum_{d|n} mu(n/d) * a(d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A343189 a(1) = ... = a(4) = 1; a(n+4) = Sum_{d|n} mu(n/d) * a(d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A343190 a(1) = ... = a(5) = 1; a(n+5) = Sum_{d|n} mu(n/d) * a(d).