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

0, 1, 1, 2, 4, 10, 24, 63, 164, 444, 1204, 3328, 9233, 25865, 72734, 205656, 583320, 1660318, 4737540, 13551165, 38837535, 111512229, 320681604, 923528963, 2663057582, 7688068638, 22218350303, 64272720521, 186091334380, 539237928902, 1563731491958, 4537823968645, 13176960639940, 38286514506439, 111306880581963

Offset

1,4

Links

Table of n, a(n) for n=1..35.

Example

a(2)=a(3)=1, because the two roots must be the (all) 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.

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,
997171512998, 2809934352700, 7929819784355, 22409533673568, 63411730258053, 179655930440464, 509588049810620, 1447023384581029,
4113254119923150, 11703780079612453, 33333125878283632] ;
g81 := add( op(i, a000081)*x^i, i=1..nops(a000081) ) ;
g81fin := x ;
g := 0 ;
nmax := nops(a000081) ;
for m from 0 to nmax do
    mhalf := floor(m/2) ;
    ghalf := g81^mhalf*g81fin ;
    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](%) ;

Crossrefs

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

Sequence in context: A138175 A121691 A124499 * A132220 A007874 A294410

Adjacent sequences: A303837 A303838 A303839 * A303841 A303842 A303843

Keyword

nonn

Author

R. J. Mathar, May 01 2018

Status

approved