A034781 -id:A034781 - cp's OEIS frontend

A000393 Number of n-node rooted trees of height 6.

0, 0, 0, 0, 0, 0, 1, 6, 26, 94, 308, 941, 2744, 7722, 21166, 56809, 149971, 390517, 1005491, 2564164, 6485901, 16289602, 40659669, 100934017, 249343899, 613286048, 1502515487, 3667953650, 8925161513, 21652815724, 52387028291

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n=1..200
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Column h=6 of A034781.

Maple
```
For Maple program see link in A000235.
```

f[n_] := Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 40}], x] &, {1}, n];f[6]-f[5] (* Geoffrey Critzer, Aug 01 2013 *)

A034823-A001385. - Christian G. Bower

A000418 Number of n-node rooted trees of height 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 7, 34, 136, 487, 1615, 5079, 15349, 45009, 128899, 362266, 1002681, 2740448, 7411408, 19865445, 52840977, 139624510, 366803313, 958696860, 2494322662, 6463281890, 16686206047, 42935345688, 110142163940, 281763465941

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n=1..200
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Column h=7 of A034781.

Maple
```
For Maple program see link in A000235.
```

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k - 1] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]]; a[n_] := b[n - 1, n - 1, 7] - b[n - 1, n - 1, 6]; Array[a, 40] (* Jean-François Alcover, Feb 08 2016, after Alois P. Heinz in A034781 *)

A034824-A034823. - Christian G. Bower

A000429 Number of n-node rooted trees of height 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 43, 188, 728, 2593, 8706, 27961, 86802, 262348, 776126, 2256418, 6466614, 18311915, 51334232, 142673720, 393611872, 1078955836, 2941029334, 7977065816, 21541492856, 57942770689, 155304829763, 414934057486

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n=1..200
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Column h=8 of A034781.

Maple
```
For Maple program see link in A000235.
```

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k - 1] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]]; a[n_] := b[n - 1, n - 1, 8] - b[n - 1, n - 1, 7]; Array[a, 40] (* Jean-François Alcover, Feb 08 2016, after Alois P. Heinz in A034781 *)

a(n) = A034825(n) - A034824(n). - Christian G. Bower

A126085 Number of n-node rooted trees of height 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 53, 251, 1043, 3961, 14102, 47816, 156129, 494769, 1530583, 4642490, 13853571, 40779959, 118666510, 341938083, 977028128, 2771407719, 7811504825, 21895173627, 61069528116, 169590519864, 469117076770

Offset: 1

N. J. A. Sloane, Mar 02 2007

nonn

N. J. A. Sloane, Table of n, a(n) for n=1..200
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Column h=9 of A034781.

Maple
```
For Maple program see link in A000235.
```

A245068 Number of n-node rooted trees of height 10.

Original entry on oeis.org

1, 10, 64, 326, 1445, 5819, 21858, 77878, 266265, 880883, 2837412, 8940811, 27661849, 84275206, 253424332, 753616430, 2219633033, 6483334299, 18800362216, 54171793605, 155219693611, 442551961032, 1256207931637, 3551733610431, 10006315323755, 28100400701241

Offset: 11

Alois P. Heinz, Jul 11 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..1000

Column h=10 of A034781.

A358554 Least Matula-Goebel number of a rooted tree with n internal (non-leaf) nodes.

Original entry on oeis.org

1, 2, 3, 5, 11, 25, 55, 121, 275, 605, 1331, 3025, 6655, 14641, 33275, 73205

Offset: 1

Gus Wiseman, Nov 27 2022

Positions of first appearances in A342507.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding rooted trees begin:
      1: o
      2: (o)
      3: ((o))
      5: (((o)))
     11: ((((o))))
     25: (((o))((o)))
     55: (((o))(((o))))
    121: ((((o)))(((o))))
    275: (((o))((o))(((o))))
    605: (((o))(((o)))(((o))))
   1331: ((((o)))(((o)))(((o))))
   3025: (((o))((o))(((o)))(((o))))
   6655: (((o))(((o)))(((o)))(((o))))
  14641: ((((o)))(((o)))(((o)))(((o))))
  33275: (((o))((o))(((o)))(((o)))(((o))))
  73205: (((o))(((o)))(((o)))(((o)))(((o))))

For height instead of internals we have A007097, firsts of A109082.
For leaves instead of internals we have A151821, firsts of A109129.
Positions of first appearances in A342507.
The ordered version gives firsts of A358553.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A001263, A206487, A358578, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[Count[MGTree[n],[_],{0,Infinity}],{n,1000}];
Table[Position[seq,n][[1,1]],{n,Union[seq]}]

A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

Original entry on oeis.org

8, 16, 24, 28, 32, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 64, 72, 76, 80, 81, 84, 96, 98, 104, 106, 108, 112, 114, 120, 126, 128, 131, 133, 136, 140, 144, 147, 148, 152, 156, 159, 160, 162, 168, 171, 172, 178, 180, 182, 184, 189, 190, 192, 196, 200, 204, 208

Offset: 1

Gus Wiseman, Dec 01 2022

nonn

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Node-height is the number of nodes in the longest path from root to leaf.

			The terms together with their corresponding rooted trees begin:
   8: (ooo)
  16: (oooo)
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  49: ((oo)(oo))
  53: ((oooo))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  57: ((o)(ooo))
  63: ((o)(o)(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))

Gus Wiseman, The first 64 rooted trees with greater width than height.

Positions of negative terms in A358726.
These trees are counted by A358728.
Differences: A358580, A358724, A358726, A358729.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A185650, A206487, A209638, A358576, A358577, A358578, A358587, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Depth[MGTree[#]]-1

A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

Original entry on oeis.org

0, 1, 0, 2, 1, 6, 7, 26, 43, 135, 276, 755, 1769, 4648, 11406, 29762, 75284, 195566, 503165, 1310705, 3402317, 8892807, 23231037, 60906456, 159786040, 420144405, 1105673058, 2914252306, 7688019511, 20304253421, 53667498236, 141976081288, 375858854594, 995728192169

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

			The a(1) = 0 through a(7) = 7 trees:
  .  (o)  .  ((oo))  ((o)(o))  (((ooo)))  (((o))(oo))
             (o(o))            ((o(oo)))  (((o)(oo)))
                               ((oo(o)))  ((o)((oo)))
                               (o((oo)))  ((o)(o(o)))
                               (o(o(o)))  ((o(o)(o)))
                               (oo((o)))  (o((o)(o)))
                                          (o(o)((o)))

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

For internals instead of leaves: A011782, ranked by A209638.
For internals instead of edge-height: A185650 aerated, ranked by A358578.
For node-height: A358589 (square trees), ranked by A358577, ordered A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internals, ordered A090181.
Cf. A065097, A109082, A109129, A342507, A358552, A358587, A358591, A358728.

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}]==Depth[#]-2&]],{n,1,10}]

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

cp's OEIS Frontend

A000393 Number of n-node rooted trees of height 6.

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A000418 Number of n-node rooted trees of height 7.

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A000429 Number of n-node rooted trees of height 8.

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Maple

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A126085 Number of n-node rooted trees of height 9.

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Maple

A245068 Number of n-node rooted trees of height 10.

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A358554 Least Matula-Goebel number of a rooted tree with n internal (non-leaf) nodes.

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Mathematica

A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

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Mathematica

A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

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