A215978 -id:A215978 - cp's OEIS frontend

A051491 Decimal expansion of Otter's rooted tree constant: lim_{n->inf} A000081(n+1)/A000081(n).

2, 9, 5, 5, 7, 6, 5, 2, 8, 5, 6, 5, 1, 9, 9, 4, 9, 7, 4, 7, 1, 4, 8, 1, 7, 5, 2, 4, 1, 2, 3, 1, 9, 4, 5, 8, 8, 3, 7, 5, 4, 9, 2, 3, 0, 4, 6, 6, 3, 5, 9, 6, 5, 9, 5, 3, 5, 0, 4, 7, 2, 4, 7, 8, 9, 0, 5, 9, 6, 4, 7, 3, 3, 1, 3, 9, 5, 7, 4, 9, 5, 1, 0, 8, 6, 6, 6, 8, 2, 8, 3, 6, 7, 6, 5, 8, 1, 3, 5, 2, 5, 3

Offset: 1

David Broadhurst

A000055(n) ~ A086308 * A051491^n * n^(-5/2), A000081(n) ~ A187770 * A051491^n * n^(-3/2). - Vaclav Kotesovec, Jan 04 2013

Analytic Combinatorics (Flajolet and Sedgewick, 2009, p. 481) has a wrong value of this constant (2.9955765856). - Vaclav Kotesovec, Jan 04 2013

			2.95576528565199497471481752412319458837549230466359659535...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.

David Broadhurst, Resurgent Integer Sequences, Rutgers Experimental Math Seminar, Feb 06 2025; Slides.
Amirmohammad Farzaneh, Mihai-Alin Badiu, and Justin P. Coon, On Random Tree Structures, Their Entropy, and Compression, arXiv:2309.09779 [cs.IT], 2023.
S. R. Finch, Otter's Tree Enumeration Constants [Broken link]
S. R. Finch, Otter's Tree Enumeration Constants [Wayback Machine]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, p. 481
Simon Plouffe, Tree-growth constant to 1800 digits
Eric Weisstein's World of Mathematics, Rooted Tree
Eric Weisstein's World of Mathematics, Tree
Index entries for sequences related to trees
Index entries for sequences related to rooted trees

Cf. A000081, A051492, A187770, A000055, A086308, A215978, A274082.

digits = 99; max = 250; s[n_, k_] := s[n, k] = a[n+1-k] + If[n < 2*k, 0, s[n-k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[k]*s[n-1, k]*k, {k, 1, n-1}]/(n-1); A[x_] := Sum[a[k]*x^k, {k, 0, max}]; eq = Log[c] == 1+Sum[A[c^-k]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; RealDigits[alpha, 10, digits] // First (* Jean-François Alcover, Sep 24 2014 *)

A215977 Number T(n,k) of simple unlabeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 3, 1, 1, 0, 4, 5, 3, 1, 1, 0, 7, 10, 6, 3, 1, 1, 0, 12, 17, 12, 6, 3, 1, 1, 0, 24, 33, 23, 13, 6, 3, 1, 1, 0, 48, 62, 47, 25, 13, 6, 3, 1, 1, 0, 107, 127, 92, 53, 26, 13, 6, 3, 1, 1, 0, 236, 267, 189, 106, 55, 26, 13, 6, 3, 1, 1

Offset: 0

Alois P. Heinz, Aug 29 2012

			T(4,1) = 3: .o-o.  .o-o.  .o-o.
            .| |.  .|  .  .|\ .
            .o-o.  .o-o.  .o o.
.
T(4,2) = 3: .o-o.  .o-o.  .o-o.
            .|/ .  .|  .  .   .
            .o o.  .o o.  .o-o.
.
T(5,1) = 4: .o-o-o.  .o-o-o.  .o-o-o.  .o-o-o.
            .|  / .  .|    .  .| |  .  . /|  .
            .o-o  .  .o-o  .  .o o  .  .o o  .
.
T(5,2) = 5: .o-o o.  .o-o o.  .o-o o.  .o o-o.  .o o-o.
            .| |  .  .|    .  .|\   .  .|\   .  .|    .
            .o-o  .  .o-o  .  .o o  .  .o-o  .  .o-o  .
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1,  1;
  0,  4,  5,  3,  1,  1;
  0,  7, 10,  6,  3,  1,  1;
  0, 12, 17, 12,  6,  3,  1,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A215981, A215982, A215983, A215984, A215985, A215986, A215987, A215988, A215989, A215980.
Row sums give: A215978.
Limiting sequence of reversed rows gives: A215979.
The labeled version of this triangle is A215861.

b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)];
g[n_] := g[n] = If[n>2, 1, 0]+b[n]-(Sum [b[k]*b[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
p[n_, i_, t_] := p[n, i, t] = If[nJean-François Alcover, Dec 04 2014, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import binomial, divisors
@cacheit
def b(n): return n if n<2 else sum([sum([d*b(d) for d in divisors(j)])*b(n - j) for j in range(1, n)])//(n - 1)
@cacheit
def g(n): return (1 if n>2 else 0) + b(n) - (sum(b(k)*b(n - k) for k in range(n + 1)) - (b(n//2) if n%2==0 else 0))//2
@cacheit
def p(n, i, t): return 0 if nIndranil Ghosh, Aug 07 2017

A144164 Number of simple graphs on n labeled nodes, where each maximally connected subgraph is either a tree or a cycle, also row sums of A144163, A215861.

Original entry on oeis.org

1, 1, 2, 8, 45, 338, 3304, 40485, 602075, 10576466, 214622874, 4941785261, 127282939615, 3625467047196, 113140481638088, 3838679644895477, 140681280613912089, 5538276165405744140, 233086092164091031114, 10443453353262112373541, 496313160155209940833001

Offset: 0

Alois P. Heinz, Sep 12 2008

nonn

			a(3) = 8, because there are 8 simple graphs on 3 labeled nodes, where each maximally connected subgraph is either a tree or a cycle, with edge-counts 0(1), 1(3), 2(3), 3(1):
.1.2. .1-2. .1.2. .1.2. .1-2. .1.2. .1-2. .1-2.
..... ..... ../.. .|... ../.. .|/.. .|... .|/..
.3... .3... .3... .3... .3... .3... .3... .3...

Alois P. Heinz, Table of n, a(n) for n = 0..140
Index entries for sequences related to trees

Row sums of triangles A144163, A215861.
The unlabeled version is A215978.
Cf. A138464, A144161, A007318, A000142.

f:= proc(n,k) option remember; local j; if k=0 then 1 elif k<0 or n<=k then 0 elif k=n-1 then n^(n-2) else add(binomial(n-1,j) *f(j+1,j) *f(n-1-j,k-j), j=0..k) fi end:
c:= proc(n,k) option remember; local i,j; if k=0 then 1 elif k<0 or n add(T(n,k), k=0..n):
seq(a(n), n=0..25);

f[n_, k_] := f[n, k] = Module[{j}, Which[k == 0, 1, k<0 || n <= k, 0, k == n-1, n^(n-2), True, Sum[Binomial[n-1, j]*f[j+1, j]*f[n-1-j, k-j], {j, 0, k}]]]; c[n_, k_] := c[n, k] = Module[{i, j}, If[k == 0, 1, If[k<0 || nJean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import binomial, prod
@cacheit
def f(n, k): return 1 if k==0 else 0 if k<0 or n<=k else n**(n - 2) if k == n - 1 else sum(binomial(n - 1, j)*f(j + 1, j)*f(n - 1 - j, k - j) for j in range(k + 1))
@cacheit
def c(n, k): return 1 if k==0 else 0 if k<0 or nIndranil Ghosh, Aug 07 2017

a(n) = Sum_{k=0..n} A144163(n,k).

a(n) ~ c * n^(n-2), where c = 1.66789780037... . - Vaclav Kotesovec, Sep 08 2014

A215981 Number of simple unlabeled graphs on n nodes with exactly 1 connected component that is a tree or a cycle.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 12, 24, 48, 107, 236, 552, 1302, 3160, 7742, 19321, 48630, 123868, 317956, 823066, 2144506, 5623757, 14828075, 39299898, 104636891, 279793451, 751065461, 2023443033, 5469566586, 14830871803, 40330829031, 109972410222, 300628862481

Offset: 1

Alois P. Heinz, Aug 29 2012

nonn

			a(5) = 4: .o-o-o.  .o-o-o.  .o-o-o.  .o-o-o.
          .|  / .  .|    .  .| |  .  . /|  .
          .o-o  .  .o-o  .  .o o  .  .o o  .

Alois P. Heinz, Table of n, a(n) for n = 1..700

Column k=1 of A215977.
The labeled version is A215851.
Cf. A000055, A215978.

with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
a:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
      (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
    end:
seq(a(n), n=1..40);

b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
a[n_] := a[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
Array[a, 40] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

a(1) = a(2) = 1, a(n) = 1 + A000055(n) for n>=3.

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A051491 Decimal expansion of Otter's rooted tree constant: lim_{n->inf} A000081(n+1)/A000081(n).

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A215977 Number T(n,k) of simple unlabeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A144164 Number of simple graphs on n labeled nodes, where each maximally connected subgraph is either a tree or a cycle, also row sums of A144163, A215861.

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A215981 Number of simple unlabeled graphs on n nodes with exactly 1 connected component that is a tree or a cycle.

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Examples

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Programs

Maple

Mathematica

Formula