A303840 -id:A303840 - cp's OEIS frontend

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

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

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Subsets of graphs in A303831. Cf. A000243 (distinguishable roots), A000055 (not rooted).
Third column of A294783.
Cf. A000081, A000107, A027852, A303840, A304067, A339303.

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

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

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

A339303 Triangle read by rows: T(n,k) is the number of unoriented linear forests with n nodes and k rooted trees.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 2, 1, 9, 6, 6, 2, 1, 20, 16, 15, 8, 3, 1, 48, 37, 41, 22, 12, 3, 1, 115, 96, 106, 69, 38, 15, 4, 1, 286, 239, 284, 194, 124, 52, 20, 4, 1, 719, 622, 750, 564, 377, 189, 77, 24, 5, 1, 1842, 1607, 2010, 1584, 1144, 618, 292, 100, 30, 5, 1

Offset: 1

Andrew Howroyd, Dec 04 2020

Linear forests (A339067) are considered up to reversal of the linear order.

T(n,k) is the number of unlabeled trees on n nodes rooted at two indistinguishable nodes at distance k-1 from each other.

			Triangle read by rows:
    1;
    1,   1;
    2,   1,   1;
    4,   3,   2,   1;
    9,   6,   6,   2,   1;
   20,  16,  15,   8,   3,   1;
   48,  37,  41,  22,  12,   3,  1;
  115,  96, 106,  69,  38,  15,  4,  1;
  286, 239, 284, 194, 124,  52, 20,  4, 1;
  719, 622, 750, 564, 377, 189, 77, 24, 5, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Columns 1..4 are A000081, A027852, A280788(n-3), A339302.
Row sums are A303840(n+2).
Row sums excluding the first column are A303833.
Cf. A339067.

\\ 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)}
ColSeq(n,k)={my(r=TreeGf(max(0,n+1-k))); Vec(r^k + r^(k%2)*subst(r, x, x^2)^(k\2), -n)/2}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

G.f of column k: (r(x)^k + r(x)^(k mod 2)*r(x^2)^floor(k/2))/2 where r(x) is the g.f. of A000081.

cp's OEIS Frontend

A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots.

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