A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots.

0, 1, 2, 6, 15, 43, 116, 329, 918, 2609, 7391, 21099, 60248, 172683, 495509, 1424937, 4102693, 11830006, 34148859, 98686001, 285459772, 826473782, 2394774727, 6944343654, 20151175658, 58513084011, 170007600051, 494230862633

Offset: 1

Author

R. J. Mathar, Brendan McKay, May 01 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Subsets of graphs in A303831. Cf. A000243 (distinguishable roots), A000055 (not rooted).

Third column of A294783.

Cf. A000081, A000107, A027852, A303840, A304067, A339303.

Programs

Maple

a000081 := [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] ;
g81 := add( op(i,a000081)*x^i,i=1..nops(a000081) ) ;
g := 0 ;
nmax := nops(a000081) ;
for m from 0 to nmax do
    mhalf := floor(m/2) ;
    ghalf := g81^(mhalf+1) ;
    gcyc := (ghalf^2+subs(x=x^2,ghalf))/2 ;
    if type(m,odd) then
        gcyc := gcyc*g81 ;
    end if;
    g := g+gcyc ;
end do:
taylor(g,x=0,nmax) ;
gfun[seriestolist](%) ; # R. J. Mathar, May 01 2018

PARI

TreeGf is A000081 as g.f.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n), t2=subst(t,x,x^2)+O(x*x^n)); Vec((2*t^2-1)/(1-t) + (1+t)/(1-t2), -n)/2} \\ Andrew Howroyd, Dec 04 2020

Formula

G.f.: [g81(x)^2/{1-g81(x)} +(1+g81(x))*g81(x^2)/{1-g81(x^2)}] /2 = [ g243(x) +(1+g81(x))*g107(x^2)]/2, where g81 is the g.f. of A000081, g243 the g.f. of A000243 and g107 the g.f. of A000107. - R. J. Mathar, May 02 2018

a(n) = A027852(n) + A304067(n). - Brendan McKay, May 05 2018

a(n) = A303840(n+2) - A000081(n). - Andrew Howroyd, Dec 04 2020