A000081 -id:A000081 - cp's OEIS frontend

A005703 Number of n-node connected graphs with at most one cycle.

1, 1, 1, 2, 4, 8, 19, 44, 112, 287, 763, 2041, 5577, 15300, 42419, 118122, 330785, 929469, 2621272, 7411706, 21010378, 59682057, 169859257, 484234165, 1382567947, 3952860475, 11315775161, 32430737380, 93044797486, 267211342954, 768096496093, 2209772802169

Offset: 0

N. J. A. Sloane, R. K. Guy

a(n) is the number of pseudotrees on n nodes. - Eric W. Weisstein, Jun 11 2012

Also unlabeled connected graphs covering n vertices with at most n edges. For this definition we have a(1) = 0 and possibly a(0) = 0. - Gus Wiseman, Feb 20 2024

			From _Gus Wiseman_, Feb 20 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 8 graphs:
  {}  .  {12}  {12,13}     {12,13,14}     {12,13,14,15}
               {12,13,23}  {12,13,24}     {12,13,14,25}
                           {12,13,14,23}  {12,13,24,35}
                           {12,13,24,34}  {12,13,14,15,23}
                                          {12,13,14,23,25}
                                          {12,13,14,23,45}
                                          {12,13,14,25,35}
                                          {12,13,24,35,45}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Washington Bomfim, Table of n, a(n) for n = 0..500
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Pseudotree
Wikipedia, Pseudoforest

Cf. A000055, A000081, A001429 (labeled A057500), A134964 (number of pseudoforests, labeled A133686).
The labeled version is A129271.
The connected complement is A140636, labeled A140638.
Non-connected: A368834 (labeled A367869) or A370316 (labeled A369191).
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A062734 counts connected graphs by number of edges.
Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, A368951, A369197, A370317, A370318.

Needs["Combinatorica`"]; nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}];
a[0] = 0;
b = Drop[Flatten[
    sol = SolveAlways[
      0 == Series[
        t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
      x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
    CoefficientList[
     Series[CycleIndex[DihedralGroup[n], s] /.
       Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
     nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; CoefficientList[
Series[r[x] - (r[x]^2 - r[x^2])/2 + d[x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 17 2014 *)

\\ TreeGf gives gf of A000081.
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)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + g(1) + (g(2) - g(1)^2)/2 + sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2)}; \\ Andrew Howroyd and Washington Bomfim, May 15 2021

a(n) = A000055(n) + A001429(n).

More terms from Vladeta Jovovic, Apr 19 2000 and from Michael Somos, Apr 26 2000

a(27) corrected and a(28) and a(29) computed by Washington Bomfim, May 14 2008

A185650 a(n) is the number of rooted trees with 2n vertices n of whom are leaves.

Original entry on oeis.org

1, 2, 8, 39, 214, 1268, 7949, 51901, 349703, 2415348, 17020341, 121939535, 885841162, 6511874216, 48359860685, 362343773669, 2736184763500, 20805175635077, 159174733727167, 1224557214545788, 9467861087020239, 73534456468877012, 573484090227222260

Offset: 1

Stepan Orevkov, Aug 29 2013

nonn

			From _Gus Wiseman_, Nov 27 2022: (Start)
The a(1) = 1 through a(3) = 8 rooted trees:
  (o)  ((oo))  (((ooo)))
       (o(o))  ((o)(oo))
               ((o(oo)))
               ((oo(o)))
               (o((oo)))
               (o(o)(o))
               (o(o(o)))
               (oo((o)))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
V. M. Kharlamov and S. Yu. Orevkov, The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves, arXiv:math/0301245 [math.AG], 2003; J. of Combinatorial Theory, Ser. A, 105 (2004), 127-142.
Index entries for sequences related to rooted trees

The ordered version is A000891, ranked by A358579.
This is the central column of A055277.
These trees are ranked by A358578.
For height = internals we have A358587.
Square trees are counted by A358589.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internals, ordered A090181.
Cf. A034781, A109129, A358580, A358581-A358584, A358591.

terms = 23;
m = 2 terms;
T[, ] = 0;
Do[T[x_, z_] = z x - x + x Exp[Sum[Series[1/k T[x^k, z^k], {x, 0, j}, {z, 0, j}], {k, 1, j}]] // Normal, {j, 1, m}];
cc = CoefficientList[#, z]& /@ CoefficientList[T[x, z] , x];
Table[cc[[2n+1, n+1]], {n, 1, terms}] (* Jean-François Alcover, Sep 14 2018 *)
art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{-2}]==n/2&]],{n,2,10,2}] (* Gus Wiseman, Nov 27 2022 *)

\\ here R is A055277 as vector of polynomials
R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
{my(A=R(2*30)); vector(#A\2, k, polcoeff(A[2*k],k))} \\ Andrew Howroyd, May 21 2018

Terms a(20) and beyond from Andrew Howroyd, May 21 2018

A087803 Number of unlabeled rooted trees with at most n nodes.

Original entry on oeis.org

1, 2, 4, 8, 17, 37, 85, 200, 486, 1205, 3047, 7813, 20299, 53272, 141083, 376464, 1011311, 2732470, 7421146, 20247374, 55469206, 152524387, 420807242, 1164532226, 3231706871, 8991343381, 25075077710, 70082143979, 196268698287, 550695545884, 1547867058882

Offset: 1

Hugo Pfoertner, Oct 12 2003

nonn

Number of equations (order conditions) that must be satisfied to achieve order n in the construction of a Runge-Kutta method for the numerical solution of an ordinary differential equation. - Hugo Pfoertner, Oct 12 2003

Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations, (1987) Wiley, Chichester
See link for more references.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
I. Th. Famelis, S. N. Papakostas and Ch. Tsitouras, Symbolic Derivation of Runge-Kutta Order Conditions.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy).
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Florian Ingels, Revisiting Tree Isomorphism: An Algorithmic Bric-à-Brac, arXiv:2309.14441 [cs.DS], 2023-2024. See p. 17.
Eric Weisstein's World of Mathematics, Rooted Tree.
Index entries for sequences related to rooted trees.

a(n) = Sum_(k=1..n) A000081(k).

Cf. A255170, A187770, A051491.

with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= proc(n) option remember; b(n) +`if`(n<1, 0, a(n-1)) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 21 2012

b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[b[n - j]* DivisorSum[j, # *b[#]&], {j, 1, n-1}]/(n-1); a[1] = 1; a[n_] := a[n] = b[n] + a[n-1]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)
t[1] = a[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n}, {m, (n-1)/k}]; a[n_] := a[n] = a[n-1] + t[n]; Table[a[n], {n, 40}] (* Vladimir Reshetnikov, Aug 12 2016 *)
Needs["NumericalDifferentialEquationAnalysis`"]
Drop[Accumulate[Join[{0},ButcherTreeCount[20]]],1] (* Peter Luschny, Aug 18 2016 *)

a000081(k) = local(A = x); if( k<1, 0, for( j=1, k-1, A /= (1 - x^j + x * O(x^k))^polcoeff(A, j)); polcoeff(A, k));
a(n) = sum(k=1, n, a000081(k)) \\ Altug Alkan, Nov 10 2015

def A087803_list(len):
    a, t = [1], [0,1]
    for n in (1..len-1):
        S = [t[n-k+1]*sum(d*t[d] for d in divisors(k)) for k in (1..n)]
        t.append(sum(S)//n)
        a.append(a[-1]+t[-1])
    return a
A087803_list(20) # Peter Luschny, Aug 18 2016

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.664861031240097088000569... . - Vaclav Kotesovec, Sep 11 2014

In the asymptotics above the constant c = A187770 / (1 - 1 / A051491). - Vladimir Reshetnikov, Aug 12 2016

Corrected and extended by Alois P. Heinz, Aug 21 2012

Renamed (old name is in comments) by Vladimir Reshetnikov, Aug 23 2016

A187770 Decimal expansion of Otter's asymptotic constant beta for the number of rooted trees.

Original entry on oeis.org

4, 3, 9, 9, 2, 4, 0, 1, 2, 5, 7, 1, 0, 2, 5, 3, 0, 4, 0, 4, 0, 9, 0, 3, 3, 9, 1, 4, 3, 4, 5, 4, 4, 7, 6, 4, 7, 9, 8, 0, 8, 5, 4, 0, 7, 9, 4, 0, 1, 1, 9, 8, 5, 7, 6, 5, 3, 4, 9, 3, 5, 4, 5, 0, 2, 2, 6, 3, 5, 4, 0, 0, 4, 2, 0, 4, 7, 6, 4, 6, 0, 5, 3, 7, 9, 8, 6

Offset: 0

Vaclav Kotesovec, Jan 04 2013

A000081(n) ~ 0.439924012571 * alpha^n * n^(-3/2), alpha = 2.95576528565199497... (see A051491)

			0.43992401257102530404090339143454476479808540794...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6., p.296
D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, p. 396.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1799, (this constant was computed by David Broadhurst in November 1999)
David Broadhurst, Resurgent Integer Sequences, Rutgers Experimental Math Seminar, Feb 06 2025; Slides.
Amirmohammad Farzaneh, Mihai-Alin Badiu, and Justin P. Coon, On Random Tree Structures, Their Entropy, and Compression, arXiv:2309.09779 [cs.IT], 2023.
Eric Weisstein's World of Mathematics, Rooted Tree

Cf. A000081, A051491, A000055, A086308.

digits = 87; max = 250; s[n_, k_] := s[n, k] = a[n+1-k] + If[n < 2*k, 0, s[n-k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[k]*s[n-1, k]*k, {k, 1, n-1}]/(n-1); A[x_] := Sum[a[k]*x^k, {k, 0, max}]; APrime[x_] := Sum[k*a[k]*x^(k-1), {k, 0, max}]; eq = Log[c] == 1 + Sum[A[c^(-k)]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; b = Sqrt[(1 + Sum[APrime[alpha^-k]/alpha^k, {k, 2, max}])/(2*Pi)]; RealDigits[b, 10, digits] // First (* Jean-François Alcover, Sep 24 2014 *)

A316651 Number of series-reduced rooted trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 12, 112, 1444, 24086, 492284, 11910790, 332827136, 10546558146, 373661603588, 14636326974270, 628032444609396, 29296137817622902, 1476092246351259964, 79889766016415899270, 4622371378514020301740, 284719443038735430679268, 18601385258191195218790756

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 12 trees:
  (1(11)), (111),
  (1(12)), (2(11)), (112),
  (1(22)), (2(12)), (122),
  (1(23)), (2(13)), (3(12)), (123).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A034691, A141268, A292504.
Cf. A316652, A316653, A316654, A316655, A316656, A319369.
Row sums of A319376.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
a:= n-> add(add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1), k=1..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 18 2018

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[Binomial[A[i, k] + j - 1, j] b[n - i*j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];
a[n_] := Sum[Sum[A[n, k-j]*(-1)^j*Binomial[k, j], {j, 0, k-1}], {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

\\ here R(n,k) is A000669, A050381, A220823, ...
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v,[0]))[n])); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

From Vaclav Kotesovec, Sep 18 2019: (Start)
a(n) ~ c * d^n * n^(n-1), where d = 1.37392076830840090205551979... and c = 0.41435722857311602982846...
a(n) ~ 2*log(2)*A326396(n)/n. (End)

Terms a(9) and beyond from Andrew Howroyd, Sep 14 2018

A005347 First differences of A005579.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 13, 20, 34, 53, 88, 143, 236, 387, 641, 1061, 1763, 2937, 4903, 8202, 13750, 23095, 38850, 65461, 110465, 186665, 315827, 535011, 907341, 1540416, 2617782, 4452846, 7581016, 12917486, 22027745, 37591270, 64196610

Offset: 0

N. J. A. Sloane, R. K. Guy, Apr 12 1988

This is example 42 in Guy's paper. a(2)-a(8) are the same as the Fibonacci sequence A000045. Subsequent terms deviate from Fibonacci. - T. D. Noe, May 08 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 0..43
Richard K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
Richard K. Guy, Letters to N. J. A. Sloane, 1986-88.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.

Cf. A000045, A005579.

prod = Interval[1]; k = k0 = 0; Join[{1, 1}, Table[While[Max[prod] <= n, k++; p = Prime[k]; prod = N[prod*p/(p - 1), 30]]; If[Min[prod] > n, If[k > 2, Print[k - k0] ]; k0 = k; k, "too few digits"], {n, 2, 39}] // Differences] (* Jean-François Alcover, Oct 07 2016, using T. D. Noe's code for A005579 *)

a(n) = A005579(n+1) - A005579(n) - T. D. Noe, May 08 2006

More terms from Harvey P. Dale, Aug 07 2013

Offset changed to 0, a(0) prepended, and a(1) inserted by Amiram Eldar, Apr 18 2025

A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 14, 35, 97, 264, 733, 2034, 5728, 16101, 45595, 129327, 368093, 1049520, 2999415, 8584857, 24612114, 70652441, 203075740, 584339171, 1683151508, 4852736072, 14003298194, 40441136815, 116880901512, 338040071375, 978314772989, 2833067885748, 8208952443400

Offset: 0

Washington Bomfim, Feb 24 2008

nonn

a(n) is the number of simple unlabeled graphs on n nodes whose components have exactly one cycle. - Geoffrey Critzer, Oct 12 2012

Also the number of unlabeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 5 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
F. Ruskey, Combinatorial Generation, p. 79, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

Cf. A008483, A137918.
The connected case is A001429.
Without the choice condition we have A001434, covering A006649.
For any number of edges we have A134964, complement A140637.
The labeled version is A137916.
The version with loops is A369145, complement A368835.
The complement is counted by A369201, labeled A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A007716, A014068, A053530, A116508, A133686, A140638, A368601, A369141, A369146.

Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]   (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n],{2}],{n}],Select[Tuples[#],UnsameQ@@#&]!={}&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

a(n) = Sum_{1*j_1 + 2*j_2 + ... = n} (Product_{i=3..n} binomial(A001429(i) + j_i -1, j_i)). [F. Ruskey p. 79, (4.27) with n replaced by n+1, and a_i replaced by A001429(i)].

Euler transform of A001429. - Geoffrey Critzer, Oct 12 2012

Edited by Washington Bomfim, Jun 27 2012
Terms a(30) and beyond from Andrew Howroyd, May 05 2018
Offset changed to 0 by Gus Wiseman, Jan 27 2024

A316473 Number of locally disjoint rooted trees with n nodes, meaning no branch overlaps any other (unequal) branch of the same root.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 44, 99, 233, 554, 1346, 3300, 8219, 20635, 52300, 133488, 343033, 886360, 2302133, 6005835

Offset: 1

Gus Wiseman, Jul 04 2018

			The a(5) = 9 locally disjoint rooted trees:
((((o))))
(((oo)))
((o(o)))
((ooo))
(o((o)))
(o(oo))
((o)(o))
(oo(o))
(oooo)

Gus Wiseman, The a(6) = 19 locally disjoint rooted trees.

Cf. A000081, A285573, A302696, A304713, A316468, A316470, A316471, A316475, A316495.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],Select[Tuples[#,2],UnsameQ@@#&&(Intersection@@#=!={})&]=={}&]];
Table[Length[strut[n]],{n,15}]

a(20) from Jinyuan Wang, Jun 20 2020

A298118 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 40, 80, 159, 322, 657, 1356, 2816, 5896, 12407, 26267, 55861, 119331, 255878, 550665, 1188786, 2574006, 5588177, 12162141, 26529873, 57993624, 127020653, 278716336, 612617523, 1348680531, 2973564157, 6565313455, 14514675376

Offset: 1

Gus Wiseman, Jan 12 2018

nonn

			The a(6) = 6 trees: (((((o))))), (((ooo))), ((oo(o))), (oo((o))), (o(o)(o)), (ooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A000598, A003238, A004111, A027193, A032305, A067659, A290689, A291443, A297791, A298120.

orut[n_]:=orut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[orut/@c]]]/@Select[IntegerPartitions[n-1],OddQ[Length[#]]&]];
Table[Length[orut[n]],{n,15}]

a(n) ~ c * d^n / n^(3/2), where d = 2.30984417428419893876754252289588812511559... and c = 0.5598122522173731208680575003383895445787... - Vaclav Kotesovec, Jun 04 2019

a(24)-a(35) from Alois P. Heinz, Jan 12 2018

A324924 Irregular triangle read by rows giving the factorization of n into factors q(i) = prime(i)/i, i > 0.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 1, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 4, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 4, 7, 1, 1, 1, 2, 2, 1, 1, 1, 8, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 2, 9

Offset: 1

Gus Wiseman, Mar 20 2019

Row n is the multiset of Matula-Goebel numbers of all proper terminal subtrees of the rooted tree with Matula-Goebel number n. For example, the rooted tree with Matula-Goebel number 1362 is (o(o)((oo)(oo))), with proper terminal subtrees {o,o,o,o,o,o,(o),(oo),(oo),((oo)(oo))}, which have Matula-Goebel numbers {1,1,1,1,1,1,2,4,4,49}, which is row 1362, as required.

			Triangle begins:
  {}
  1
  1  2
  1  1
  1  2  3
  1  1  2
  1  1  4
  1  1  1
  1  1  2  2
  1  1  2  3
  1  2  3  5
  1  1  1  2
  1  1  2  6
  1  1  1  4
  1  1  2  2  3
  1  1  1  1
  1  1  4  7
  1  1  1  2  2
  1  1  1  8
  1  1  1  2  3
  1  1  1  2  4
  1  1  2  3  5
  1  1  2  2  9
For example, row 65 is {1,1,1,2,2,3,6} because 65 = q(1)^3 q(2)^2 q(3) q(6).

Cf. A000081, A003963, A061775, A109082, A109129, A112798, A120383, A196050, A317713, A324850, A324922, A324923, A324925, A324931, A324934.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[difac[n],{n,30}]

cp's OEIS Frontend

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A316651 Number of series-reduced rooted trees with n leaves spanning an initial interval of positive integers.

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A005347 First differences of A005579.

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A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

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