A000107 -id:A000107 - cp's OEIS frontend

A038002 Number of connected functions on n points with a single labeled point.

0, 1, 3, 9, 27, 81, 242, 722, 2150, 6395, 19003, 56428, 167458, 496724, 1472835, 4365692, 12936998, 38327764, 113529027, 336221554, 995586119, 2947641940, 8726093434, 25829729702, 76450357119, 226257478851, 669566448376, 1981320898874, 5862583555761

Offset: 0

Christian G. Bower

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000107, A027853, A339067.

G.f.: B(x)(B(x)+1) where B(x) is g.f. of A000107.
a(n) = Sum_{k=1..n} k * A339067(n,k). - Alois P. Heinz, Dec 04 2020
G.f.: A000081(x) / (1 - A000081(x))^2, where A000081(x) is the g.f. of A000081. - Vaclav Kotesovec, Jan 03 2021

A214567 Maximal number of distinct rooted trees obtained from the rooted tree with Matula-Goebel number n by adding one pendant edge at one of its vertices.

Original entry on oeis.org

1, 2, 3, 2, 4, 4, 3, 2, 3, 5, 5, 4, 5, 4, 6, 2, 4, 4, 3, 5, 5, 6, 4, 4, 4, 6, 3, 4, 6, 7, 6, 2, 7, 5, 6, 4, 5, 4, 7, 5, 6, 6, 5, 6, 6, 5, 7, 4, 3, 5, 6, 6, 3, 4, 8, 4, 5, 7, 5, 7, 5, 7, 5, 2, 8, 8, 4, 5, 6, 7, 6, 4, 6, 6, 6, 4, 7, 8, 7, 5, 3

Offset: 1

Emeric Deutsch, Jul 25 2012

nonn

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

Sum_{j such that rooted tree with Matula-Goebel number j has n vertices} a(j) = A000107(n). Example: the Matula-Goebel numbers of the rooted trees with 4 vertices are 5,6,7,8 and a(5)+a(6)+a(7)+a(8) = 4+4+3+2=13 = A000107(4).

			a(4)=2 because the rooted tree with Matula-Goebel number 4 is V; adding an edge at either of the two leaves yields the same rooted tree.
a(5)=4 because the rooted tree with Matula-Goebel number 5 is the path on 4 vertices; adding one edge at any of the vertices yields a new rooted tree.
a(987654321)=18 (reader may verify this on Fig. 2 of the Deutsch paper).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A000107.

Cf. A049084, A027748.

import Data.List (genericIndex)
a214567 n = genericIndex a214567_list (n - 1)
a214567_list = 1 : g 2 where
  g x = y : g (x + 1) where
    y | t > 0     = a214567 t + 1
      | otherwise = 1 + sum (map ((subtract 1) . a214567) $ a027748_row x)
       where t = a049084 x
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local FS: FS := proc (n) options operator, arrow: factorset(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+a(pi(n)) else 1+add(a(FS(n)[j])-1, j = 1 .. nops(FS(n))) end if end proc: seq(a(n), n = 1 .. 130);

a[n_] := Which[n == 1, 1, PrimeQ[n], 1 + a[PrimePi[n]], True, 1 + Total[a[#] - 1& /@ FactorInteger[n][[All, 1]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 25 2024 *)

a(n) = 1 + vecsum([self()(primepi(p)) |p<-factor(n)[,1]]); \\ Kevin Ryde, Oct 19 2022

a(1)=1; if n is t-th prime, then a(n)=1+a(t); if n is composite, then a(n) = 1+Sum_{p|n}(a(p)-1), where summation is over the distinct prime divisors of n.

A303911 Triangle T(w>=1,1<=n<=w) read by rows: the number of rooted weighted trees with n nodes and weight w.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 10, 13, 9, 1, 5, 16, 31, 35, 20, 1, 6, 24, 60, 98, 95, 48, 1, 7, 33, 103, 217, 304, 262, 115, 1, 8, 44, 162, 423, 764, 945, 727, 286, 1, 9, 56, 241, 743, 1658, 2643, 2916, 2033, 719, 1, 10, 70, 341, 1221, 3224, 6319, 8996, 8984, 5714, 1842, 1, 11, 85, 466, 1893

Offset: 1

R. J. Mathar, May 02 2018

Weights are positive integer labels on the nodes. The weight of the tree is the sum of the weights of its nodes.

			The triangle starts
1 ;
1  1 ;
1  2  2 ;
1  3  5   4 ;
1  4 10  13    9 ;
1  5 16  31   35    20 ;
1  6 24  60   98    95    48 ;
1  7 33 103  217   304   262   115 ;
The first column (for a single node n=1) is 1, because all the weight is on that node.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math. 101 (1959) 141-162, W(x,y) equation (9a)

Cf. A000081 (diagonal), A000107 (subdiagonal), A036249 (row sums), A303841 (not rooted).

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerMT(y*v))); v}
{my(A=seq(10)); for(n=1, #A, print(Vecrev(A[n])))} \\ Andrew Howroyd, May 19 2018

A051529 INVERT transform of A000081 = [1, 1, 1, 2, 4, 9, 20, 48, 115, 286,...].

Original entry on oeis.org

1, 2, 4, 9, 21, 51, 126, 318, 812, 2100, 5482, 14438, 38303, 102302, 274824, 742210, 2013941, 5488239, 15014376, 41221775, 113542455, 313681756, 868994723, 2413526848, 6719132105, 18746838609, 52412080624, 146812972155, 411977724704, 1158006098132

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..650

Cf. A000081, A000107.

with(numtheory): b:= proc(n) option remember; local d, j; `if`(n<=1, 1, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= proc(n) option remember; local i; `if`(n<1, 1, add(a(n-i) *b(i-1), i=1..n+1)) end: seq(a(n), n=0..30);  # Alois P. Heinz, Apr 01 2009

b[n_] := b[n] = If[n <= 1, 1, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/ (n - 1)];
a[n_] := a[n] = If[n < 1, 1, Sum[a[n - i]*b[i - 1], {i, 1, n + 1}]];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 01 2021, after Alois P. Heinz *)

A262253 A weighted sum over the rooted trees of n nodes (A214568).

Original entry on oeis.org

0, 1, 3, 9, 29, 90, 285, 886, 2764, 8543, 26387, 81091, 248752, 760687, 2321950, 7072376

Offset: 0

R. J. Mathar, Sep 16 2015

(More precise name desired.)

R. Harary, R. W. Robinson, Isomorphic factorizations VIII: bisectable trees, Combinatorica 4 (2) (1984) 169-179, function F(x).
Index entries for sequences related to rooted trees

Cf. A000107, A000081, A007098.

a(n) = sum_{k>=1} binomial(k+1,2) A214568(n,k).
A007098(x) = A(x) -A(x^2) -A000081(x)*A(x) -{A000107(x)^2 - A000107(x^2)}/2 is the relation between the generating functions, eq. prior to (4.9) by Harary-Robinson.
A(x) = A000081(x)*{A(x)-A(x^2)+ A000107(x^2)/2} +{A000081(x)+A000107(x)+A000107(x)^2}/2 , eq. (4.6) by Harary-Robinson.

A280785 Triangle read by rows: numbers of nonintersecting circles, one marked.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 2, 1, 13, 13, 6, 2, 1, 35, 35, 16, 6, 2, 1, 95, 95, 46, 17, 6, 2, 1, 262, 262, 128, 49, 17, 6, 2, 1, 727, 727, 364, 139, 50, 17, 6, 2, 1, 2033, 2033, 1029, 401, 142, 50, 17, 6, 2, 1, 5714, 5714, 2930, 1147, 412, 143, 50, 17, 6, 2, 1, 16136, 16136, 8344, 3299, 1184, 415, 143, 50, 17, 6, 2, 1

Offset: 1

N. J. A. Sloane, Jan 20 2017

			Triangle begins:
1,
1,1,
2,2,1,
5,5,2,1,
13,13,6,2,1,
35,35,16,6,2,1,
95,95,46,17,6,2,1,
262,262,128,49,17,6,2,1,
727,727,364,139,50,17,6,2,1,
...

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Row sums are A000107.

cp's OEIS Frontend

A038002 Number of connected functions on n points with a single labeled point.

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A214567 Maximal number of distinct rooted trees obtained from the rooted tree with Matula-Goebel number n by adding one pendant edge at one of its vertices.

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A303911 Triangle T(w>=1,1<=n<=w) read by rows: the number of rooted weighted trees with n nodes and weight w.

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PARI

A051529 INVERT transform of A000081 = [1, 1, 1, 2, 4, 9, 20, 48, 115, 286,...].

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Maple

Mathematica

A262253 A weighted sum over the rooted trees of n nodes (A214568).

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Formula

A280785 Triangle read by rows: numbers of nonintersecting circles, one marked.

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