A244925 -id:A244925 - cp's OEIS frontend

A048815 Number of rooted trees with n nodes with every leaf at height 10.

1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 145, 245, 425, 724, 1246, 2130, 3659, 6254, 10724, 18335, 31396, 53676, 91832, 156944, 268324, 458435, 783324, 1337862, 2284950, 3901211, 6660304, 11367935, 19401130, 33104598, 56481086, 96349147

Offset: 11

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=10 of A244925.

Euler transform of A048814 shifted right.

A048810 Number of rooted trees with n nodes with every leaf at height 5.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 26, 44, 73, 123, 203, 340, 563, 935, 1550, 2571, 4251, 7034, 11618, 19188, 31654, 52201, 85999, 141631, 233074, 383375, 630215, 1035508, 1700501, 2791309, 4579587, 7510280, 12310980, 20172075, 33039130, 54092556

Offset: 6

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=5 of A244925.

Euler transform of A048809 shifted right.

A048811 Number of rooted trees with n nodes with every leaf at height 6.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 27, 47, 78, 135, 224, 384, 642, 1088, 1827, 3088, 5182, 8736, 14661, 24660, 41378, 69500, 116534, 195509, 327627, 549104, 919593, 1539985, 2577399, 4313102, 7214374, 12064930, 20169283, 33710370, 56324729, 94089240

Offset: 7

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=6 of A244925.

Euler transform of A048810 shifted right.

A048812 Number of rooted trees with n nodes with every leaf at height 7.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 48, 81, 140, 236, 405, 686, 1169, 1984, 3375, 5723, 9721, 16478, 27949, 47354, 80245, 135869, 230054, 389304, 658706, 1114072, 1883900, 3184602, 5382321, 9094154, 15362767, 25946131, 43811971, 73964065

Offset: 8

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=7 of A244925.

Euler transform of A048811 shifted right.

A048813 Number of rooted trees with n nodes with every leaf at height 8.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 49, 82, 143, 241, 417, 707, 1213, 2065, 3534, 6014, 10272, 17487, 29820, 50758, 86469, 147123, 250429, 425932, 724517, 1231765, 2094116, 3558799, 6047447, 10273349, 17450221, 29633832, 50317376, 85420630

Offset: 9

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=8 of A244925.

Euler transform of A048812 shifted right.

A048814 Number of rooted trees with n nodes with every leaf at height 9.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 144, 244, 422, 719, 1234, 2109, 3615, 6173, 10565, 18042, 30839, 52653, 89927, 153462, 261931, 446818, 762190, 1299678, 2215990, 3777230, 6437673, 10969447, 18688879, 31834676, 54220089, 92331502

Offset: 10

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=9 of A244925.

Euler transform of A048813 shifted right.

A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

Original entry on oeis.org

0, 2, 10, 12, 42, 52, 56, 170, 204, 212, 232, 240, 682, 820, 844, 852, 920, 936, 976, 992, 2730, 3276, 3284, 3380, 3404, 3412, 3640, 3688, 3736, 3752, 3888, 3920, 4000, 4032, 10922, 13108, 13132, 13140, 13516, 13524, 13620, 13644, 13652, 14568, 14744, 14760

Offset: 1

Gus Wiseman, Nov 21 2022

nonn

An ordered tree is balanced if all leaves are the same distance from the root.

The binary encoding of an ordered tree (see A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

			The terms together with their corresponding trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  204: ((o)(o))
  212: ((ooo))
  232: (((oo)))
  240: ((((o))))
  682: (ooooo)
  820: ((o)(oo))
  844: ((oo)(o))
  852: ((oooo))
  920: (((o)(o)))
  936: (((ooo)))
  976: ((((oo))))
  992: (((((o)))))

These trees are counted by A007059.
This is a subset of A014486.
The version for binary trees is A057122.
The unordered version is A184155, counted by A048816.
Another ranking of balanced ordered trees is A358459.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A001263, A003238, A244925, A358377, A358379, A358523.

binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]]
Select[Range[0,1000],binbalQ[#]&&SameQ@@Length/@Position[bint[#],{}]&]

A074045 Number of rooted trees of 2n+1 nodes with every leaf at height n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 145, 246, 427, 729, 1256, 2152, 3702, 6341, 10892, 18662, 32016, 54853, 94034, 161055, 275929, 472461, 809033, 1384848, 2370434, 4056309, 6940744, 11873769, 20311018, 34737781, 59405959, 101577454, 173669088, 296890268

Offset: 0

Christian G. Bower, Aug 13 2002. Suggested by Paul D. Hanna

Vaclav Kotesovec, Table of n, a(n) for n = 0..3500 (terms 0..250 from Alois P. Heinz)
Index entries for sequences related to rooted trees

Limit of A048808-A048815.

Cf. A244925.

a(n) = A244925(2n+1,n).

A320221 Irregular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k, (n>=1, min(1,n-1) <= k <= log_2(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 7, 1, 1, 11, 4, 1, 13, 6, 1, 20, 16, 1, 23, 23, 1, 33, 46, 1, 40, 70, 1, 54, 127, 1, 1, 65, 189, 1, 1, 87, 320, 5, 1, 104, 476, 10, 1, 136, 771, 32, 1, 164, 1145, 63, 1, 209, 1795, 154, 1, 252, 2657, 304, 1, 319, 4091, 656

Offset: 1

Gus Wiseman, Oct 07 2018

			Triangle begins:
  1
  1
  1
  1  1
  1  1
  1  3
  1  3
  1  6  1
  1  7  1
  1 11  4
  1 13  6
  1 20 16
  1 23 23
  1 33 46
  1 40 70
The T(11,3) = 6 rooted trees:
   (((oo)(oo))((oo)(ooooo)))
   (((oo)(oo))((ooo)(oooo)))
   (((oo)(ooo))((oo)(oooo)))
   (((oo)(ooo))((ooo)(ooo)))
  (((oo)(oo))((oo)(oo)(ooo)))
  (((oo)(ooo))((oo)(oo)(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..1154 (rows 1..200)

Row sums are A120803. Second column is A083751. A regular version is A320179.
Cf. A000669, A005804, A048816, A119262, A120803, A141268, A244925, A319312.
Cf. A316624, A320154, A320155, A320160, A320172, A320173.

qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
DeleteCases[Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,10},{k,0,n-1}],0,{2}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
T(n)={my(u=vector(n), v=vector(n), h=1); u[1]=1; while(u, v+=u*h; h*=x; u=EulerT(u)-u); v[1]=x; [Vecrev(p/x) | p<-v]}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020

Terms a(36) and beyond from Andrew Howroyd, Dec 09 2020

Name clarified by Andrew Howroyd, Dec 09 2020

A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 35, 37, 41, 43, 64, 128, 129, 137, 139, 163, 169, 171, 256, 257, 293, 512, 515, 529, 547, 553, 555, 641, 649, 651, 675, 681, 683, 1024, 1025, 2048, 2053, 2057, 2059, 2177, 2185, 2187, 2211, 2217, 2219, 2305, 2341, 2563

Offset: 1

Gus Wiseman, Nov 19 2022

nonn

An ordered tree is balanced if all leaves have the same distance from the root.

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  16: (oooo)
  17: ((((o))))
  32: (ooooo)
  35: ((oo)(o))
  37: (((o))((o)))
  41: ((o)(oo))
  43: ((o)(o)(o))

Gus Wiseman, Statistics, classes, and transformations of standard compositions

These trees are counted by A007059.
The unordered version is A184155, counted by A048816.
A000108 counts ordered rooted trees, unordered A000081.
A358379 gives depth of standard ordered trees.
Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],SameQ@@Length/@Position[srt[#],{}]&]

cp's OEIS Frontend

A048815 Number of rooted trees with n nodes with every leaf at height 10.

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A048810 Number of rooted trees with n nodes with every leaf at height 5.

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A048811 Number of rooted trees with n nodes with every leaf at height 6.

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A048812 Number of rooted trees with n nodes with every leaf at height 7.

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A048813 Number of rooted trees with n nodes with every leaf at height 8.

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A048814 Number of rooted trees with n nodes with every leaf at height 9.

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A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

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Mathematica

A074045 Number of rooted trees of 2n+1 nodes with every leaf at height n.

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A320221 Irregular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k, (n>=1, min(1,n-1) <= k <= log_2(n)).

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PARI

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A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

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Mathematica