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A303833
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Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots.
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8
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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
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OFFSET
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1,3
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..1000
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FORMULA
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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
a(n) = A303840(n+2) - A000081(n). - Andrew Howroyd, Dec 04 2020
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MAPLE
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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
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PROG
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(PARI) TreeGf is A000081 as g.f.
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)}
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
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CROSSREFS
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Subsets of graphs in A303831. Cf. A000243 (distinguishable roots), A000055 (not rooted).
Third column of A294783.
Cf. A000081, A000107, A027852, A303840, A304067, A339303.
Sequence in context: A221744 A338861 A340726 * A148438 A148439 A151515
Adjacent sequences: A303830 A303831 A303832 * A303834 A303835 A303836
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KEYWORD
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nonn
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AUTHOR
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R. J. Mathar, Brendan McKay, May 01 2018
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STATUS
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approved
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