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 A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots. 8
 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 Andrew Howroyd, Table of n, a(n) for n = 1..1000 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 a(n) = A303840(n+2) - A000081(n). - Andrew Howroyd, Dec 04 2020 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 PROG (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 CROSSREFS 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: A178936 A221744 A338861 * 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 January 15 21:50 EST 2021. Contains 340195 sequences. (Running on oeis4.)