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A027852 Number of connected functions on n points with a loop of length 2. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also rooted trees on n+1 nodes where root has degree 2.

Also trees on n nodes in which one pair of adjacent nodes is labeled in a special way. - Washington Bomfim, Jul 06 2012

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

Second column of A033185. - Michael Somos, Aug 20 2018

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

FORMULA

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

a(n) = Sum_(p=1..L-1) {f(p) * f(n-p)} + f(L) * f(n-L) * (n mod 2) + (f(L)+1)*f(L)/2*(1-n mod 2), where L = floor(n/2), and f = A000081. - Washington Bomfim, Jul 06 2012

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

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-Fran├žois Alcover, Oct 30 2018, after Alois P. Heinz *)

PROG

(PARI)

max_n = 650; f=vector(max_n);           \\ f[n] = A000081[n], n=1..max_n

sum2(k) = {local(s); s=0; fordiv(k, d, s += d*f[d]); return(s)};

Init_f()={f[1]=1;

for(n =1, max_n -2, s=0; for(k=1, n, s+=sum2(k)*f[n-k+1]); f[n+1]=s/n)};

a(n) = {s=0; if(n==1, return(s)); L = floor(n/2);

for(p=1, L-1, s += f[p]*f[n-p]);

s += f[L]*f[n-L]*(n%2) + (f[L]+1)*f[L]/2*(1-n%2); return(s)}

Init_f(); for(n=1, max_n, print(n, " ", a(n))) \\ b-file format

\\ Washington Bomfim, Jul 06 2012

CROSSREFS

Cf. A000081, A000226, A001372, A002861, A033185, A051491.

Sequence in context: A293993 A072824 A089406 * A203068 A321229 A114410

Adjacent sequences:  A027849 A027850 A027851 * A027853 A027854 A027855

KEYWORD

nonn

AUTHOR

Christian G. Bower, Dec 14 1997

EXTENSIONS

Edited by Christian G. Bower, Feb 12 2002

STATUS

approved

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Last modified January 20 16:20 EST 2019. Contains 319335 sequences. (Running on oeis4.)