A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots.
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
Offset: 1
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1000
Crossrefs
Programs
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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 -
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
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