A007562 -id:A007562 - cp's OEIS frontend

A319381 Number of plane trees with n nodes where the sequence of branches directly under any given node is a membership-chain.

1, 1, 1, 2, 2, 4, 6, 9, 11, 20, 28, 40, 58, 82, 110, 159, 217, 305, 420, 570, 767, 1042

Offset: 1

Gus Wiseman, Sep 17 2018

			The a(9) = 11 membership-chain trees:
  ((((((((o))))))))  (((((((o)o))))))  ((((((o)o)o))))  (((((o)o)o)o))
                     ((((((o))(o)))))  (((((o)o)(o))))  ((((o)o)(o)o))
                     (((((o)))((o))))  (((((o))(o)o)))  ((((o))(o)o)o)
                                       ((((o))(o))(o))

Gus Wiseman, The a(15) = 110 membership-chain trees.

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A316470, A319122, A319379, A319380, A319436.

yanplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[yanplane/@c],And@@MemberQ@@@Partition[#,2,1]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[yanplane[n]],{n,10}]

A319378 Number of plane trees with n nodes where the sequence of branches directly under any given node with at least two branches has empty intersection.

Original entry on oeis.org

1, 1, 2, 5, 13, 39, 118, 375, 1225, 4079, 13794, 47287, 163962, 573717, 2023800

Offset: 1

Gus Wiseman, Sep 17 2018

			The a(5) = 13 locally nonintersecting plane trees:
  ((((o))))  (((oo)))  ((ooo))  (oooo)
             (((o)o))  ((oo)o)
             ((o(o)))  (o(oo))
             (((o))o)  ((o)oo)
             (o((o)))  (o(o)o)
                       (oo(o))

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A143363, A316470, A319122, A319379, A319380, A319381, A319436.

monplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[monplane/@c],Or[Length[#]==1,Intersection@@#=={}]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[monplane[n]],{n,10}]

A363387 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / (k*x^k) ).

Original entry on oeis.org

1, 1, 1, 3, 6, 17, 42, 120, 330, 962, 2797, 8334, 24989, 75905, 232142, 715830, 2220473, 6928411, 21723883, 68424327, 216376757, 686742855, 2186771571, 6984248840, 22368127861, 71818903891, 231132440916, 745454242656, 2409080380316, 7799945417349

Offset: 1

Ilya Gutkovskiy, May 30 2023

nonn

Cf. A005750, A007562, A363388.

nmax = 30; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[x^k]^2/(k x^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 + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]

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, subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A319271 Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 3, 9, 12, 27, 42, 91, 151, 312, 550, 1099, 2026, 3999, 7527, 14804, 28336, 55641, 107737, 211851, 413508, 814971, 1600512, 3162761, 6241234

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches, and aperiodic if the multiplicities in the multiset of branches directly under any given node are relatively prime, and locally non-intersecting if the branches directly under any given node with more than one branch have empty intersection.

			The a(8) = 9 rooted trees:
  (o(o(o(o))))
  (o(o(o)(o)))
  (o(ooo(o)))
  (oo(oo(o)))
  (o(o)(o(o)))
  (ooo(o(o)))
  (o(o)(o)(o))
  (ooo(o)(o))
  (ooooo(o))

Cf. A000081, A000837, A007562, A289509, A301700, A303431, A316470, A316473, A316475, A316495, A319270.

btrut[n_]:=btrut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[btrut/@c]]]/@IntegerPartitions[n-1],And[Intersection@@#=={},GCD@@Length/@Split[#]==1]&]];
Table[Length[btrut[n]],{n,30}]

A345200 G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} A(x^k) / k).

Original entry on oeis.org

1, 1, 2, 5, 12, 32, 84, 234, 652, 1872, 5416, 15922, 47188, 141283, 425910, 1293105, 3948080, 12118619, 37367694, 115708111, 359623780, 1121543440, 3508533500, 11006973980, 34620982004, 109157354769, 344928572562, 1092190467567, 3464955417200, 11012117992012

Offset: 1

Ilya Gutkovskiy, Jun 11 2021

nonn

Cf. A000151, A007562, A345241, A345242.

nmax = 30; A[] = 0; Do[A[x] = x + x^2 Exp[2 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] = (2/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]

G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(2*a(n)).
a(n+2) = (2/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 3.3437762102302517833309792925121217026126033230718263962128740290952197... and c = 0.3397354606156870289877990463189432389789387070060129709272911771... - Vaclav Kotesovec, Jun 19 2021

A345234 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, 2, 3, 5, 9, 17, 31, 58, 112, 218, 427, 844, 1683, 3381, 6824, 13842, 28226, 57796, 118762, 244874, 506515, 1050688, 2185095, 4555217, 9517423, 19926174, 41798031, 87833877, 184881588, 389765182, 822901122, 1739763655, 3682955618, 7806103024, 16564348106, 35187631009

Offset: 1

Ilya Gutkovskiy, Jun 11 2021

nonn

Cf. A007560, A007562, A049075, A345235.

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

G.f.: x + x^2 * Product_{n>=1} (1 + (-x)^n)^((-1)^n*a(n)).
a(n+2) = (1/n) * Sum_{k=1..n} (-1)^(k+1) * ( Sum_{d|k} (-1)^(k/d+d) * d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 2.21094707842288180828190718521597733363607957468229824761... and c = 0.664585976397397791197984310778764361056468131968... - Vaclav Kotesovec, Jun 19 2021

A345235 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, 3, 5, 8, 14, 25, 44, 78, 142, 261, 479, 886, 1655, 3105, 5843, 11043, 20965, 39938, 76285, 146123, 280691, 540475, 1042885, 2016481, 3906647, 7582034, 14739395, 28697969, 55958110, 109262713, 213619535, 418158580, 819491034, 1607764395, 3157551026, 6207346544

Offset: 1

Ilya Gutkovskiy, Jun 11 2021

nonn

Cf. A007560, A007562, A045648, A345234.

nmax = 40; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[(-1)^(k + 1) 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[(-1)^k Sum[(-1)^d d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 40}]

G.f.: x + x^2 / Product_{n>=1} (1 - (-x)^n)^((-1)^n*a(n)).
a(n+2) = (1/n) * Sum_{k=1..n} (-1)^k * ( Sum_{d|k} (-1)^d * d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 2.04187801797233390910633071122033289228232310618876458... and c = 0.624667034123125135463988884805660643637934291759335... - Vaclav Kotesovec, Jun 19 2021

A345241 G.f. A(x) satisfies: A(x) = x + x^2 * exp(3 * Sum_{k>=1} A(x^k) / k).

Original entry on oeis.org

1, 1, 3, 9, 28, 93, 315, 1109, 3969, 14505, 53726, 201588, 764001, 2921730, 11257881, 43669590, 170383933, 668236581, 2632898016, 10416893159, 41368099791, 164841324837, 658883345595, 2641064296638, 10613953319448, 42757746556377, 172628891937513, 698398635475974

Offset: 1

Ilya Gutkovskiy, Jun 11 2021

nonn

Cf. A006964, A007562, A345200, A345242.

nmax = 28; A[] = 0; Do[A[x] = x + x^2 Exp[3 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] = (3/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]

G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(3*a(n)).

a(n+2) = (3/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).

A345242 G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum_{k>=1} A(x^k) / k).

Original entry on oeis.org

1, 1, 4, 14, 52, 205, 832, 3492, 14960, 65322, 289384, 1298064, 5882712, 26897352, 123919576, 574718308, 2681028168, 12571650355, 59222213028, 280139215118, 1330101884932, 6336757979653, 30282375754944, 145124083402256, 697293746743760, 3358385599930269, 16210842955175380

Offset: 1

Ilya Gutkovskiy, Jun 11 2021

nonn

Cf. A007562, A052763, A345200, A345241.

nmax = 27; A[] = 0; Do[A[x] = x + x^2 Exp[4 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] = (4/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 27}]

G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(4*a(n)).

a(n+2) = (4/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).

A345973 G.f.: x + x^2 / Product_{n>=1} (1 - a(n)*x^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 36, 73, 138, 281, 549, 1136, 2263, 4705, 9553, 20015, 41096, 86643, 179638, 380701, 795892, 1693003, 3562217, 7612680, 16099538, 34505797, 73345831, 157678081, 336419942, 725236780, 1552662599, 3354979195, 7205601904, 15600414855, 33594465666

Offset: 1

Ilya Gutkovskiy, Jun 30 2021

nonn

David Callan, A Combinatorial Interpretation for Sequence A345973 in OEIS, arXiv:2108.04969 [math.CO], 2021.

Cf. A007562, A032307, A093637.

a:= proc(n) option remember; `if`(n<3, 1, add(a(n-k)*add(d*
      a(d)^(k/d), d=numtheory[divisors](k)), k=1..n-2)/(n-2))
    end:
seq(a(n), n=1..37);  # Alois P. Heinz, Jul 01 2021

a[n_] := a[n] = SeriesCoefficient[x + x^2/Product[(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 37}]
a[1] = a[2] = 1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 37}]

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

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

cp's OEIS Frontend

A319381 Number of plane trees with n nodes where the sequence of branches directly under any given node is a membership-chain.

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A319378 Number of plane trees with n nodes where the sequence of branches directly under any given node with at least two branches has empty intersection.

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A363387 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / (k*x^k) ).

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A319271 Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.

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A345200 G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} A(x^k) / k).

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A345234 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 + ...).

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A345235 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 + ...).

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A345241 G.f. A(x) satisfies: A(x) = x + x^2 * exp(3 * Sum_{k>=1} A(x^k) / k).

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A345242 G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum_{k>=1} A(x^k) / k).

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A345973 G.f.: x + x^2 / Product_{n>=1} (1 - a(n)*x^n).

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