cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A173765 Partial sums of A005200.

Original entry on oeis.org

1, 3, 7, 18, 46, 124, 337, 935, 2605, 7328, 20684, 58670, 166863, 476032, 1360955, 3899324, 11191494, 32173714, 92625281, 267010344, 770610783, 2226438062, 6438902174, 18638275524, 53995855636, 156548609636, 454200201824
Offset: 1

Views

Author

Jonathan Vos Post, Feb 24 2010

Keywords

Comments

x. The subsequence of primes in this partial sum begins: 3, 7, 337, 121208284196421093429539251 = a(58).

Examples

			a(28) = 1 + 2 + 4 + 11 + 28 + 78 + 213 + 598 + 1670 + 4723 + 13356 + 37986 + 108193 + 309169 + 884923 + 2538369 + 7292170 + 20982220 + 60451567 + 174385063 + 503600439 + 1455827279 + 4212464112 + 12199373350 + 35357580112 + 102552754000 + 297651592188 + 86446068277.

Crossrefs

Cf. A005200, A000081, A005201, A000055.

Formula

a(n) = SUM[i=1..n] A005200(i).

A000081 Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point).

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228, 35221832, 97055181, 268282855, 743724984, 2067174645, 5759636510, 16083734329, 45007066269, 126186554308, 354426847597, 997171512998
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, number of ways of arranging n-1 nonoverlapping circles: e.g., there are 4 ways to arrange 3 circles, as represented by ((O)), (OO), (O)O, OOO, also see example. (Of course the rules here are different from the usual counting parentheses problems - compare A000108, A001190, A001699.) See Sloane's link for a proof and Vogeler's link for illustration of a(7) as arrangement of 6 circles.
Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible legal ways (cf. A003018). Sequence gives number of distinct functions. The single node tree is "x". Making a node f2 a child of f1 represents f1^f2. Since (f1^f2)^f3 is just f1^(f2*f3) we can think of it as f1 raised to both f2 and f3, that is, f1 with f2 and f3 as children. E.g., for n=4 the distinct functions are ((x^x)^x)^x; (x^(x^x))^x; x^((x^x)^x); x^(x^(x^x)). - W. Edwin Clark and Russ Cox, Apr 29 2003; corrected by Keith Briggs, Nov 14 2005
Also, number of connected multigraphs of order n without cycles except for one loop. - Washington Bomfim, Sep 04 2010
Also, number of planted trees with n+1 nodes.
Also called "Polya trees" by Genitrini (2016). - N. J. A. Sloane, Mar 24 2017

Examples

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 48*x^7 + 115*x^8 + ...
From _Joerg Arndt_, Jun 29 2014: (Start)
The a(6) = 20 trees with 6 nodes have the following level sequences (with level of root = 0) and parenthesis words:
  01:  [ 0 1 2 3 4 5 ]    (((((())))))
  02:  [ 0 1 2 3 4 4 ]    ((((()()))))
  03:  [ 0 1 2 3 4 3 ]    ((((())())))
  04:  [ 0 1 2 3 4 2 ]    ((((()))()))
  05:  [ 0 1 2 3 4 1 ]    ((((())))())
  06:  [ 0 1 2 3 3 3 ]    (((()()())))
  07:  [ 0 1 2 3 3 2 ]    (((()())()))
  08:  [ 0 1 2 3 3 1 ]    (((()()))())
  09:  [ 0 1 2 3 2 3 ]    (((())(())))
  10:  [ 0 1 2 3 2 2 ]    (((())()()))
  11:  [ 0 1 2 3 2 1 ]    (((())())())
  12:  [ 0 1 2 3 1 2 ]    (((()))(()))
  13:  [ 0 1 2 3 1 1 ]    (((()))()())
  14:  [ 0 1 2 2 2 2 ]    ((()()()()))
  15:  [ 0 1 2 2 2 1 ]    ((()()())())
  16:  [ 0 1 2 2 1 2 ]    ((()())(()))
  17:  [ 0 1 2 2 1 1 ]    ((()())()())
  18:  [ 0 1 2 1 2 1 ]    ((())(())())
  19:  [ 0 1 2 1 1 1 ]    ((())()()())
  20:  [ 0 1 1 1 1 1 ]    (()()()()())
(End)

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
  • N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, pp. 42, 49.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 305, 998.
  • A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).
  • J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.
  • F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 54 and 244.
  • Alexander S. Karpenko, Łukasiewicz Logics and Prime Numbers, Luniver Press, Beckington, 2006, p. 82.
  • D. E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3d Ed. 1997, pp. 386-388.
  • D. E. Knuth, The Art of Computer Programming, vol. 1, 3rd ed., Fundamental Algorithms, p. 395, ex. 2.
  • D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6.
  • G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987, p. 63.
  • R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998. [Comment from Neven Juric: Page 64 incorrectly gives a(21)=35224832.]
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000041 (partitions), A000055 (unrooted trees), A000169, A001858, A005200, A027750, A051491, A051492, A093637, A187770, A199812, A255170, A087803 (partial sums).
Row sums of A144963. - Gary W. Adamson, Sep 27 2008
Cf. A209397 (log(A(x)/x)).
Cf. A000106 (self-convolution), A002861 (rings of these).
Column k=1 of A033185 and A034799; column k=0 of A008295.

Programs

  • Haskell
    import Data.List (genericIndex)
a000081 = genericIndex a000081_list
a000081_list = 0 : 1 : f 1 [1,0] where
   f x ys = y : f (x + 1) (y : ys) where
     y = sum (zipWith (*) (map h [1..x]) ys) `div` x
     h = sum . map (\d -> d * a000081 d) . a027750_row
-- Reinhard Zumkeller, Jun 17 2013
  • Magma
    N := 30; P := PowerSeriesRing(Rationals(),N+1); f := func< A | x*&*[Exp(Evaluate(A,x^k)/k) : k in [1..N]]>; G := x; for i in [1..N] do G := f(G); end for; G000081 := G; A000081 := [0] cat Eltseq(G); // Geoff Bailey (geoff(AT)maths.usyd.edu.au), Nov 30 2009
  • Maple
    N := 30: a := [1,1]; for n from 3 to N do x*mul( (1-x^i)^(-a[i]), i=1..n-1); series(%,x,n+1); b := coeff(%,x,n); a := [op(a),b]; od: a; A000081 := proc(n) if n=0 then 1 else a[n]; fi; end; G000081 := series(add(a[i]*x^i,i=1..N),x,N+2); # also used in A000055
spec := [ T, {T=Prod(Z,Set(T))} ]; A000081 := n-> combstruct[count](spec,size=n); [seq(combstruct[count](spec,size=n), n=0..40)];
# a much more efficient method for computing the result with Maple. It uses two procedures:
a := proc(n) local k; a(n) := add(k*a(k)*s(n-1,k), k=1..n-1)/(n-1) end proc:
a(0) := 0: a(1) := 1: s := proc(n,k) local j; s(n,k) := add(a(n+1-j*k), j=1..iquo(n,k)); # Joe Riel (joer(AT)san.rr.com), Jun 23 2008
# even more efficient, uses the Euler transform:
with(numtheory): a:= proc(n) option remember; local d, j; `if`(n<=1, n, (add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-1))/ (n-1)) end:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 06 2008
  • Mathematica
    s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] (* Robert A. Russell *)
a[n_] := a[n] = If[n <= 1, n, Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-1}]/(n-1)]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
a[n_] := a[n] = If[n <= 1, n, Sum[a[n - j] DivisorSum[j, # a[#] &], {j, n - 1}]/(n - 1)]; Table[a[n], {n, 0, 30}] (* Jan Mangaldan, May 07 2014, after Alois P. Heinz *)
(* first do *) << NumericalDifferentialEquationAnalysis`; (* then *)
ButcherTreeCount[30] (* v8 onward Robert G. Wilson v, Sep 16 2014 *)
a[n:0|1] := n; a[n_] := a[n] = Sum[m a[m] a[n-k*m], {m, n-1}, {k, (n-1)/m}]/(n-1); Table[a[n], {n, 0, 30}] (* Vladimir Reshetnikov, Nov 06 2015 *)
terms = 31; A[] = 0; Do[A[x] = x*Exp[Sum[A[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 11 2018 *)
  • Maxima
    g(m):= block([si,v],s:0,v:divisors(m), for si in v do (s:s+r(m/si)/si),s);
r(n):=if n=1 then 1 else sum(Co(n-1,k)/k!,k,1,n-1);
Co(n,k):=if k=1  then g(n)  else sum(g(i+1)*Co(n-i-1,k-1),i,0,n-k);
makelist(r(n),n,1,12); /*Vladimir Kruchinin, Jun 15 2012 */
  • PARI
    {a(n) = local(A = x); if( n<1, 0, for( k=1, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff(A, n))}; /* Michael Somos, Dec 16 2002 */
  • PARI
    {a(n) = local(A, A1, an, i); if( n<1, 0, an = Vec(A = A1 = 1 + O(x^n)); for( m=2, n, i=m\2; an[m] = sum( k=1, i, an[k] * an[m-k]) + polcoeff( if( m%2, A *= (A1 - x^i)^-an[i], A), m-1)); an[n])}; /* Michael Somos, Sep 05 2003 */
  • PARI
    N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1,n, sumdiv(k,d, d*A[d]) * A[n-k+1] ) );
concat([0], A) \\ Joerg Arndt, Apr 17 2014
  • Python
    from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def divisor_tuple(n): # cached unordered tuple of divisors
    return tuple(divisors(n,generator=True))
@lru_cache(maxsize=None)
def A000081(n): return n if n <= 1 else sum(sum(d*A000081(d) for d in divisor_tuple(k))*A000081(n-k) for k in range(1,n))//(n-1) # Chai Wah Wu, Jan 14 2022
  • Sage
    @CachedFunction
def a(n):
    if n < 2: return n
    return add(add(d*a(d) for d in divisors(j))*a(n-j) for j in (1..n-1))/(n-1)
[a(n) for n in range(31)] # Peter Luschny, Jul 18 2014 after Alois P. Heinz
  • Sage
    [0]+[RootedTrees(n).cardinality() for n in range(1,31)] # Freddy Barrera, Apr 07 2019

Formula

G.f. A(x) satisfies A(x) = x*exp(A(x)+A(x^2)/2+A(x^3)/3+A(x^4)/4+...) [Polya]
Also A(x) = Sum_{n>=1} a(n)*x^n = x / Product_{n>=1} (1-x^n)^a(n).
Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d*a(d) ) * a(n-k+1).
Asymptotically c * d^n * n^(-3/2), where c = A187770 = 0.439924... and d = A051491 = 2.955765... [Polya; Knuth, section 7.2.1.6].
Euler transform is sequence itself with offset -1. - Michael Somos, Dec 16 2001
For n > 1, a(n) = A087803(n) - A087803(n-1). - Vladimir Reshetnikov, Nov 06 2015
For n > 1, a(n) = A123467(n-1). - Falk Hüffner, Nov 26 2015

A005201 Total number of fixed points in trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 5, 10, 31, 72, 201, 509, 1374, 3587, 9647, 25686, 69348, 187052, 508480, 1384959, 3791466, 10407842, 28677319, 79231664, 219557624, 609922977, 1698526750, 4740469708, 13258136509, 37151664771, 104294992317, 293279485007
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A005200, A000081, A000055.

Programs

  • Maple
    # First form T(x) = g.f. for A000081 and F(x) = g.f. for A005200. Then:
t1 := subs(x=x^2,F); series(T*(1-t1)-t1, x,31);
# second Maple program:
with(numtheory): t:= proc(n) option remember; local d, j; if n<1 then 0 elif n=1 then 1 else add(add(d*t(d), d=divisors(j)) *t(n-j), j=1..n-1)/ (n-1) fi end: f:= proc(n) option remember; t(n) +add((t(n-i) -t(n-2*i)) *f(i), i=0..n-1) end: t1 := n-> `if`(type(n,odd), 0,f(n/2)): a:= proc(n) t(n) -add(t(n-i) *t1(i), i=0..n) -t1(n) end: seq(a(n), n=1..50); # Alois P. Heinz, Sep 17 2008
  • Mathematica
    t[n_] := t[n] = If[n<1, 0, If[n == 1, 1, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]]; f[n_] := f[n] = t[n]+Sum[(t[n-i]-t[n-2*i])*f[i], {i, 0, n-1}]; t1[n_] := If[OddQ[n], 0, f[n/2]]; a[n_] := t[n]-Sum[t[n-i]*t1[i], {i, 0, n}]-t1[n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)

Formula

G.f. satisfies A(x) = T(x)*(1-F(x^2))-F(x^2), where T(x) = x + x^2 + 2*x^3 + ... is g.f. for A000081, F(x) = x + 2*x^2 + 4*x^3 + 11*x^4 + ... is the g.f. for A005200.

A005202 Total number of fixed points in planted trees with n nodes.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 6, 14, 28, 60, 125, 263, 558, 1181, 2513, 5339, 11392, 24290, 51926, 111017, 237757, 509404, 1092713, 2345256, 5038015, 10828720, 23291759, 50126055, 107939753, 232550011, 501270200, 1080996244, 2332221316, 5033764628, 10868950676, 23476998980, 50728408182, 109649040738, 237081174662, 512767906801, 1109354495908
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From R. J. Mathar, Apr 13 2019: (Start)
The associated triangle H_{p,j}, p >= 1, 1 <= j <= p, a(n) = Sum_{j=1..p} j*H_{p,j}, row sums in A001678, starts:
1;
0, 0;
1, 0, 0;
1, 0, 0, 0;
1, 0, 1, 0, 0;
1, 1, 1, 0, 0, 0;
2, 2, 1, 0, 1, 0, 0;
1, 4, 2, 2, 1, 0, 0, 0;
3, 4, 4, 5, 2, 0, 1, 0, 0;
3, 7, 7, 9, 4, 4, 1, 0, 0, 0;
5, 9, 15, 14, 11, 9, 3, 0, 1, 0, 0;
4, 14, 23, 28, 25, 19, 7, 6, 1, 0, 0, 0;
11, 15, 39, 46, 55, 38, 24, 14, 5, 0, 1, 0, 0;
6, 32, 54, 86, 97, 86, 64, 36, 11, 9, 1, 0, 0, 0;
(End)

References

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

Links

Crossrefs

Cf. A005200.

Programs

  • Maple
    Hpj := proc(Hofxy,p,j)
    coeftayl(Hofxy,x=0,p) ;
    coeftayl(%,y=0,j) ;
    simplify(%) ;
end proc:
Hxy := proc(x,y,pmax,hxyinit)
    if pmax = 0 then
        x*y ;
    else
        pp := 1;
        for p from 1 to pmax do
            t :=1 ;
            for j from 1 to p do
                t := t*(1+x^p*y^j+add(x^(k*p),k=2..pmax+1))^Hpj(hxyinit,p,j) ;
            end do:
            pp := pp*t ;
        end do:
        x*y*%/(1+x*y) ;
    end if;
end proc:
hxy := Hxy(x,y,0,0) ;
for pmax from 2 to 20 do
    Hxy(x,y,pmax,hxy) ;
    taylor(%,x=0,pmax+2) ;
    convert(%,polynom) ;
    taylor(%,y=0,pmax+2) ;
    hxy := convert(%,polynom) ;
    for p from 0 to pmax do
        Ap := 0 ;
        for j from 1 to p do
            Ap := Ap+j*Hpj(hxy,p,j) ;
        end do:
        printf("%d,",Ap) ;
    end do:
    print() ;
end do: # R. J. Mathar, Apr 13 2019
  • Mathematica
    Hpj[Hofxy_, p_, j_] := SeriesCoefficient[SeriesCoefficient[Hofxy, {x, 0, p}] , {y, 0, j}];
Hxy [x_, y_, pMax_, hxyinit_] := If [pMax == 0, x y, pp = 1; For[p = 1, p <= pMax, p++, t = 1; For[j = 1, j <= p, j++, t = t(1 + x^p y^j + Sum[x^(k*p), {k, 2, pMax + 1}])^Hpj[hxyinit, p, j]]; pp = pp t]; x*y* pp/(1 + x y)];
hxy = Hxy[x, y, 0, 0];
Reap[For[pMax = 2, pMax <= terms - 1, pMax++, Print["pMax = ", pMax]; sx = Series[Hxy[x, y, pMax, hxy], {x, 0, pMax + 2}] // Normal; sy = Series[sx, {y, 0, pMax + 2}]; hxy = sy // Normal; For[p = 0, p <= pMax, p++, Ap = 0; For[j = 1, j <= p, j++, Ap = Ap + j Hpj[hxy, p, j]]; If[pMax == terms - 1, Print[Ap]; Sow[Ap]]]]][[2, 1]] (* Jean-François Alcover, Mar 22 2020, after R. J. Mathar *)

Extensions

More terms from R. J. Mathar, Apr 13 2019

A185667 Primes in A000081.

Original entry on oeis.org

2, 719, 87811
Offset: 1

Views

Author

Jonathan Vos Post, Feb 08 2011

Keywords

Comments

A000081(n) is prime for n in A051420.
a(4) = A000081(343) has 158 digits and is too large to be displayed here.

Links

Crossrefs

Cf. A000040, A000081, A005200, A051420, A119641, A119642.

Formula

A000040 INTERSECTION A000081.
a(n) = A000081(A051420(n)). - Amiram Eldar, Nov 11 2017

A037246 Total number of fixed points in free homeomorphically irreducible trees with n nodes.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 5, 10, 16, 38, 66, 143, 268, 564, 1100, 2282, 4546, 9382, 18977, 39112, 79891, 164917, 339195, 702041, 1451628, 3013442, 6257561, 13029327, 27152492, 56698062, 118518363, 248137778, 520085704, 1091520783, 2293229235, 4823466463
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A005200-A005202.

Programs

  • Maple
    Hpj := proc(Hofxy,p,j)
    coeftayl(Hofxy,x=0,p) ;
    coeftayl(%,y=0,j) ;
    simplify(%) ;
end proc:
Hxy := proc(x,y,pmax,hxyinit)
    if pmax = 0 then
        x*y ;
    else
        pp := 1;
        for p from 1 to pmax do
            t :=1 ;
            for j from 1 to p do
                t := t*(1+x^p*y^j+add(x^(k*p),k=2..pmax+1))^Hpj(hxyinit,p,j) ;
            end do:
            pp := pp*t ;
        end do:
        x*y*%/(1+x*y) ;
    end if;
end proc:
hxyfin := Hxy(x,y,0,0) ;
for pmax from 2 to 40 do
    Hxy(x,y,pmax,hxyfin) ;
    taylor(%,x=0,pmax+2) ;
    convert(%,polynom) ;
    taylor(%,y=0,pmax+2) ;
    hxyfin := convert(%,polynom) ;
    hxy := (1+x*y)*hxyfin+subs({x=x^2,y=1},hxyfin)*(1-x*y)-hxyfin^2*(1+x*y)/2+subs({x=x^2,y=y^2},hxyfin)*(x*y-1)/2 ;
    for p from 0 to pmax do
        ap := 0 ;
        for j from 1 to p do
            ap := ap+j*Hpj(hxy,p,j) ;
        end do:
        printf("%d,",ap) ;
    end do:
    print() ;
end do: # R. J. Mathar, Apr 13 2019

Formula

Reference gives a recurrence.

Extensions

More terms from R. J. Mathar, Apr 13 2019

A051496 Decimal expansion of the probability that a point of an infinite (rooted) tree is fixed by every automorphism of the tree.

Original entry on oeis.org

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

Views

Author

David Broadhurst

Keywords

Comments

F. Harary and E. M. Palmer derive certain functional equations and, using the methods of G. Polya (Acta Math. (1937) Vol. 68, 145-254) and R. Otter (Ann. of Math. (2) 49 (1948), 583-599; Math. Rev. 10, 53), prove that the limiting probability of a fixed point in a large random tree, whether rooted or not, is 0.6995...

Examples

			0.6995388700609892332166312186...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.6.3, p. 304.

Links

Crossrefs

Cf. A000055, A000081, A005200, A005201.

Formula

Equals lim_{n->oo} A005200(n)/(n*A000081(n)).
Equals lim_{n->oo} A005201(n)/(n*A000055(n)).
Showing 1-7 of 7 results.

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