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A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..28.

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

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

CROSSREFS

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

Third column of A294783.

Sequence in context: A065178 A178936 A221744 * A148438 A148439 A151515

Adjacent sequences:  A303830 A303831 A303832 * A303834 A303835 A303836

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified July 24 03:24 EDT 2019. Contains 325290 sequences. (Running on oeis4.)