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

%I A303833 #27 Dec 04 2020 17:23:21
%S A303833 0,1,2,6,15,43,116,329,918,2609,7391,21099,60248,172683,495509,
%T A303833 1424937,4102693,11830006,34148859,98686001,285459772,826473782,
%U A303833 2394774727,6944343654,20151175658,58513084011,170007600051,494230862633
%N A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots.
%H A303833 Andrew Howroyd, <a href="/A303833/b303833.txt">Table of n, a(n) for n = 1..1000</a>
%F A303833 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
%F A303833 a(n) = A027852(n) + A304067(n). - _Brendan McKay_, May 05 2018
%F A303833 a(n) = A303840(n+2) - A000081(n). - _Andrew Howroyd_, Dec 04 2020
%p A303833 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] ;
%p A303833 g81 := add( op(i,a000081)*x^i,i=1..nops(a000081) ) ;
%p A303833 g := 0 ;
%p A303833 nmax := nops(a000081) ;
%p A303833 for m from 0 to nmax do
%p A303833     mhalf := floor(m/2) ;
%p A303833     ghalf := g81^(mhalf+1) ;
%p A303833     gcyc := (ghalf^2+subs(x=x^2,ghalf))/2 ;
%p A303833     if type(m,odd) then
%p A303833         gcyc := gcyc*g81 ;
%p A303833     end if;
%p A303833     g := g+gcyc ;
%p A303833 end do:
%p A303833 taylor(g,x=0,nmax) ;
%p A303833 gfun[seriestolist](%) ; # _R. J. Mathar_, May 01 2018
%o A303833 (PARI) TreeGf is A000081 as g.f.
%o A303833 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)}
%o A303833 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
%Y A303833 Subsets of graphs in A303831. Cf. A000243 (distinguishable roots), A000055 (not rooted).
%Y A303833 Third column of A294783.
%Y A303833 Cf. A000081, A000107, A027852, A303840, A304067, A339303.
%K A303833 nonn
%O A303833 1,3
%A A303833 _R. J. Mathar_, _Brendan McKay_, May 01 2018