A027852 Number of connected functions on n points with a loop of length 2.

0, 1, 1, 3, 6, 16, 37, 96, 239, 622, 1607, 4235, 11185, 29862, 80070, 216176, 586218, 1597578, 4370721, 12003882, 33077327, 91433267, 253454781, 704429853, 1962537755, 5479855546, 15332668869, 42983656210, 120716987723, 339596063606, 956840683968

Offset: 1

Author

Christian G. Bower, Dec 14 1997

Keywords

• nonn

Comments

Number of unordered pairs of rooted trees with a total of n nodes.

Equivalently, the number of rooted trees on n+1 nodes where the root has degree 2.

Number of trees on n nodes rooted at an edge. - Washington Bomfim, Jul 06 2012

Guy (1988) calls these tadpole graphs. - N. J. A. Sloane, Nov 04 2014

Number of unicyclic graphs of n nodes with a cycle length of two (in other words, a double edge). - Washington Bomfim, Dec 02 2020

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2136
• Washington Bomfim, Illustration of initial terms
• R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6.
• R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Eq. (75).
• Index entries for sequences related to rooted trees

Crossrefs

Column 2 of A033185 (forests of rooted trees), A217781 (unicyclic graphs), A339303 (unoriented linear forests) and A339428 (connected functions).

Cf. A000081, A000106, A000226, A000631, A001372, A002861.

Programs

Maple

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:= n-> (add(b(i) *b(n-i), i=0..n) +`if`(irem(n, 2)=0, b(n/2), 0))/2: seq(a(n), n=1..50);  # Alois P. Heinz, Aug 22 2008, revised Oct 07 2011
# second, re-usable version
A027852 := proc(N::integer)
    local dh, Nprime;
    dh := 0 ;
    for Nprime from 0 to N do
        dh := dh+A000081(Nprime)*A000081(N-Nprime) ;
    end do:
    if type(N,'even') then
        dh := dh+A000081(N/2) ;
    end if;
    dh/2 ;
end proc: # R. J. Mathar, Mar 06 2017

Mathematica

Needs["Combinatorica`"];nn = 30; 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[i] s[n - 1, i] i, {i, 1, n - 1}]/(n - 1); rt = Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[DihedralGroup[2], s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {2, nn}]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
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_] := (Sum[b[i] b[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, b[n/2], 0])/2;
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 30 2018, after Alois P. Heinz *)

PARI

seq(max_n)= { my(V = f = vector(max_n), i=1,s); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n = 1, max_n, s = sum(k = 1, (n-1)/2, ( f[k] * f[n-k] ));
if(n % 2 == 1, V[i] = s, V[i] = s + (f[n/2]^2 + f[n/2])/2); i++); V };
\\ Washington Bomfim, Jul 06 2012 and Dec 01 2020

Formula

G.f.: A(x) = (B(x)^2 + B(x^2))/2 where B(x) is g.f. of A000081.

a(n) = Sum_{k=1..(n-1)/2}( f(k)*f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2+f(n/2) ) /2, where f(n) = A000081(n). - Washington Bomfim, Jul 06 2012 and Dec 01 2020

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.43992401257102530404090339... . - Vaclav Kotesovec, Sep 12 2014

2*a(n) = A000106(n) + A000081(n/2), where A(.)=0 if the argument is non-integer. - R. J. Mathar, Jun 04 2020

Extensions

Edited by Christian G. Bower, Feb 12 2002