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A303840 Unlabeled trees with n nodes rooted at 2 indistinguishable roots that are leaves.

%I A303840 #14 Jun 25 2024 01:30:10
%S A303840 0,1,1,2,4,10,24,63,164,444,1204,3328,9233,25865,72734,205656,583320,
%T A303840 1660318,4737540,13551165,38837535,111512229,320681604,923528963,
%U A303840 2663057582,7688068638,22218350303,64272720521,186091334380,539237928902,1563731491958,4537823968645,13176960639940,38286514506439,111306880581963
%N A303840 Unlabeled trees with n nodes rooted at 2 indistinguishable roots that are leaves.
%e A303840 a(2)=a(3)=1, because the two roots must be (all) the leaves. a(4)=2 (one pattern from the linear tree, one from the star tree). a(6)=10: 1 pattern from n-Hexane. 2 patterns from 2-Methyl-Pentane. 2 patterns from (2,3)-Bimethyl-Butane. 1 pattern from the star graph. 2 patterns from 3-Methyl-Pentane. 2 patterns from (2,2)-Bimethyl-Butane.
%p A303840 a000081 := [1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228,
%p A303840 35221832, 97055181, 268282855, 743724984, 2067174645, 5759636510, 16083734329, 45007066269, 126186554308, 354426847597,
%p A303840 997171512998, 2809934352700, 7929819784355, 22409533673568, 63411730258053, 179655930440464, 509588049810620, 1447023384581029,
%p A303840 4113254119923150, 11703780079612453, 33333125878283632] ;
%p A303840 g81 := add( op(i,a000081)*x^i,i=1..nops(a000081) ) ;
%p A303840 g81fin := x ;
%p A303840 g := 0 ;
%p A303840 nmax := nops(a000081) ;
%p A303840 for m from 0 to nmax do
%p A303840     mhalf := floor(m/2) ;
%p A303840     ghalf := g81^mhalf*g81fin ;
%p A303840     gcyc := (ghalf^2+subs(x=x^2,ghalf))/2 ;
%p A303840     if type(m,odd) then
%p A303840         gcyc := gcyc*g81 ;
%p A303840     end if;
%p A303840     g := g+gcyc ;
%p A303840 end do:
%p A303840 taylor(g,x=0,nmax) ;
%p A303840 gfun[seriestolist](%) ;
%Y A303840 Cf. A303833 (roots need not be leaves), A055290 (cardinality of candidates).
%K A303840 nonn
%O A303840 1,4
%A A303840 _R. J. Mathar_, May 01 2018