A007562 -id:A007562 - cp's OEIS frontend

A346020 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

1, 0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 46, 74, 124, 206, 343, 577, 976, 1649, 2808, 4792, 8200, 14073, 24228, 41782, 72246, 125164, 217262, 377784, 658072, 1148006, 2005743, 3509125, 6147422, 10782375, 18934209, 33285291, 58575080, 103181405, 181928014, 321059155

Offset: 1

Ilya Gutkovskiy, Jul 01 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..3800

Cf. A007562, A218020, A346030.

a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*
       add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Jul 01 2021

nmax = 42; A[] = 0; Do[A[x] = x + x^3 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = 0; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 42}]

G.f.: x + x^3 / Product_{n>=1} (1 - x^n)^a(n).
a(1) = 1, a(2) = 0, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} d * a(d) ) * a(n-k).
a(n) ~ c * d^n / n^(3/2), where d = 1.82975393308934955558864748939303527364978309460948926333116466766295641... and c = 0.8335864368398390652263577663136791087027831725508605623969711758177... - Vaclav Kotesovec, Jul 06 2021

A358460 Number of locally disjoint ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 103, 301, 902, 2767, 8637, 27324, 87409, 282319, 919352

Offset: 1

Gus Wiseman, Nov 19 2022

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex.

			The a(1) = 1 through a(5) = 13 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (o(o))   ((ooo))
                 (((o)))  (o(o)o)
                          (o(oo))
                          (oo(o))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o(o)))
                          (o((o)))
                          ((((o))))

The locally non-intersecting version is A143363, unordered A007562.
The unordered version is A316473, ranked by A316495.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A006013, A302696, A316471, A316694, A318185, A319378, A324768, A324844.

aot[n_]:=If[n==1,{{}},Join @@ Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,{_,x_,_},_,{_,x_,_},_}]&]],{n,10}]

A363465 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^3 / (kx^(2k)) ).

Original entry on oeis.org

1, 1, 1, 4, 10, 35, 113, 405, 1447, 5369, 20143, 76908, 296800, 1157784, 4554142, 18050308, 72003513, 288880549, 1164867528, 4718481975, 19190711729, 78338352168, 320851617424, 1318115448886, 5430133003281, 22427330328214, 92847100210382, 385217596191075, 1601483701650310

Offset: 1

Ilya Gutkovskiy, Jun 03 2023

nonn

Cf. A007562, A052751, A363387, A363466, A363467.

nmax = 29; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[x^k]^3/(k x^(2 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[a[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d + 2], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 29}]

A363466 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^4 / (kx^(3k)) ).

Original entry on oeis.org

1, 1, 1, 5, 15, 61, 240, 1019, 4387, 19462, 87649, 401077, 1856698, 8685295, 40978465, 194806667, 932141498, 4486014160, 21699575863, 105443142514, 514469464550, 2519437043753, 12379461876092, 61013509071216, 301553269618318, 1494229881209940, 7421627743464582, 36942997716584746

Offset: 1

Ilya Gutkovskiy, Jun 03 2023

nonn

Cf. A007562, A052773, A363387, A363465, A363468.

nmax = 28; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[x^k]^4/(k x^(3 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[f[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d + 3], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]

A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 67, 573, 6933, 97147, 1666999

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other branch of the same root.

			The a(3) = 9 trees:
  (1(11))
   (111)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
Examples of rooted trees that are not locally disjoint are ((11)(12)) and ((12)(13)).

Cf. A000081, A007562, A316473, A316475, A316494, A316495, A316496, A316497.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
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_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

A345232 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Original entry on oeis.org

1, 1, -1, -1, 2, 0, -4, 3, 7, -12, -8, 35, -6, -87, 84, 172, -367, -187, 1175, -417, -3003, 3621, 5723, -15126, -4374, 47813, -26192, -119731, 175835, 211797, -699210, -57982, 2148031, -1601079, -5161935, 9125489, 8093890, -34478125, 3997517, 101971205, -97182026

Offset: 1

Ilya Gutkovskiy, Jun 11 2021

sign

Cf. A007562, A045648, A345233.

a:= proc(n) option remember; `if`(n<3, 1, -add(add(a(n-k)*
      d*a(d), d=numtheory[divisors](k)), k=1..n-2)/(n-2))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Jun 11 2021

nmax = 41; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[-A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; a[n_] := a[n] = -(1/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 41}]

G.f.: x + x^2 * Product_{n>=1} (1 - x^n)^a(n).

a(n+2) = -(1/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).

A346030 G.f. A(x) satisfies: A(x) = x^2 + x^3 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 2, 3, 3, 6, 8, 11, 18, 26, 37, 60, 87, 132, 206, 310, 475, 742, 1130, 1759, 2737, 4236, 6618, 10348, 16139, 25350, 39767, 62456, 98401, 155047, 244570, 386639, 611298, 967874, 1534297, 2433584, 3864154, 6141560, 9766908, 15547187, 24766037, 39476846

Offset: 1

Ilya Gutkovskiy, Jul 01 2021

nonn

Cf. A007562, A218020, A346020.

a:= proc(n) option remember; `if`(n<4, signum(n-1), add(a(n-k)*
       add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..47);  # Alois P. Heinz, Jul 01 2021

nmax = 47; A[] = 0; Do[A[x] = x^2 + x^3 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 0; a[2] = 1; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 47}]

G.f.: x^2 + x^3 / Product_{n>=1} (1 - x^n)^a(n).
a(1) = 0, a(2) = 1, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} d * a(d) ) * a(n-k).
a(n) ~ c * d^n / n^(3/2), where d = 1.646504994482771446591056040381099740295861136174688956979834656... and c = 0.8402317368556115946120005582458627329843217960728964299829... - Vaclav Kotesovec, Jul 06 2021

A363062 G.f. A(x) satisfies: A(x) = x - x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Original entry on oeis.org

1, -1, -1, 0, 1, 1, -1, -2, 0, 4, 4, -5, -13, -2, 26, 30, -29, -94, -26, 189, 246, -198, -769, -302, 1512, 2228, -1372, -6691, -3425, 12672, 21046, -9503, -60776, -38353, 109719, 205330, -61001, -567518, -427145, 967914, 2045196, -314417, -5405209, -4743873, 8625547

Offset: 1

Ilya Gutkovskiy, May 16 2023

sign

Cf. A007562, A045648, A345235.

nmax = 45; A[] = 0; Do[A[x] = x - x^2 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = -1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 45}]

G.f.: x - x^2 / Product_{n>=1} (1 - x^n)^a(n).

a(1) = 1, a(2) = -1; a(n) = (1/(n - 2)) * Sum_{k=1..n-2} ( Sum_{d|k} d * a(d) ) * a(n-k).

A363385 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / k ).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 4, 11, 14, 29, 66, 115, 222, 493, 944, 1884, 4020, 8175, 16618, 35198, 73220, 151844, 321036, 676778, 1421828, 3016813, 6407344, 13589888, 28962702, 61853827, 132073646, 282752030, 606492428, 1301587833, 2797816706, 6023460551, 12978238202, 27995493484

Offset: 1

Ilya Gutkovskiy, May 30 2023

nonn

Cf. A005750, A007562, A363386.

nmax = 38; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[x^k]^2/k, {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]

seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(sum(k=1, m\2, subst(p + O(x^(m\k+1)), x, x^k)^2/k))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A319291 Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 12, 107, 1299, 20764, 412957, 9817743

Offset: 1

Gus Wiseman, Sep 16 2018

			The a(3) = 12 series-reduced locally disjoint rooted trees:
  (1(11))
   (111)
  (1(22))
  (2(12))
   (122)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
The trees counted by A316651(4) but not by a(4):
  ((11)(12))
  ((12)(13))
  ((12)(22))
  ((12)(23))
  ((13)(23))

Cf. A000081, A007562, A301700, A316473, A316475, A316495, A316651, A316694, A316695, A316696, A316697, A319286.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
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_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
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}]

cp's OEIS Frontend

A346020 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A358460 Number of locally disjoint ordered rooted trees with n nodes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363465 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^3 / (k*x^(2*k)) ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A363466 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^4 / (k*x^(3*k)) ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A345232 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A346030 G.f. A(x) satisfies: A(x) = x^2 + x^3 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A363062 G.f. A(x) satisfies: A(x) = x - x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A363385 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / k ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A319291 Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.

Views

Author

A363465 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^3 / (kx^(2k)) ).

A363466 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^4 / (kx^(3k)) ).