A303840 - OEIS

%I #14 Jun 25 2024 01:30:10

%S 0,1,1,2,4,10,24,63,164,444,1204,3328,9233,25865,72734,205656,583320,

%T 1660318,4737540,13551165,38837535,111512229,320681604,923528963,

%U 2663057582,7688068638,22218350303,64272720521,186091334380,539237928902,1563731491958,4537823968645,13176960639940,38286514506439,111306880581963

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

%e 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 a000081 := [1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228,

%p 35221832, 97055181, 268282855, 743724984, 2067174645, 5759636510, 16083734329, 45007066269, 126186554308, 354426847597,

%p 997171512998, 2809934352700, 7929819784355, 22409533673568, 63411730258053, 179655930440464, 509588049810620, 1447023384581029,

%p 4113254119923150, 11703780079612453, 33333125878283632] ;

%p g81 := add( op(i,a000081)*x^i,i=1..nops(a000081) ) ;

%p g81fin := x ;

%p g := 0 ;

%p nmax := nops(a000081) ;

%p for m from 0 to nmax do

%p     mhalf := floor(m/2) ;

%p     ghalf := g81^mhalf*g81fin ;

%p     gcyc := (ghalf^2+subs(x=x^2,ghalf))/2 ;

%p     if type(m,odd) then

%p         gcyc := gcyc*g81 ;

%p     end if;

%p     g := g+gcyc ;

%p end do:

%p taylor(g,x=0,nmax) ;

%p gfun[seriestolist](%) ;

%Y Cf. A303833 (roots need not be leaves), A055290 (cardinality of candidates).

%K nonn

%O 1,4

%A _R. J. Mathar_, May 01 2018