A000081 -id:A000081 - cp's OEIS frontend

A216649 Triangle T(n,k) in which n-th row lists in increasing order all positive integers with a representation as totally balanced 2n digit binary string such that all consecutive totally balanced substrings are in nondecreasing order; n>=1, 1<=k<=A000081(n+1).

2, 10, 12, 42, 44, 52, 56, 170, 172, 180, 184, 204, 212, 216, 232, 240, 682, 684, 692, 696, 716, 724, 728, 744, 752, 820, 824, 852, 856, 872, 880, 920, 936, 944, 976, 992, 2730, 2732, 2740, 2744, 2764, 2772, 2776, 2792, 2800, 2868, 2872, 2900, 2904, 2920, 2928

Offset: 1

Alois P. Heinz, Sep 12 2012

There is a simple bijection between the elements of row n and the rooted trees with n+1 nodes. The tree has a root node. Each matching pair (1,0) in the binary string representation encodes an additional node, the totally balanced substrings encode lists of subtrees.

			172 is element of row 4, the binary string representation (with totally balanced substrings enclosed in parentheses) is (10)(10)(1(10)0).  The encoded rooted tree is:
.    o
.   /|\
.  o o o
.      |
.      o
Triangle T(n,k) begins:
2;
10,     12;
42,     44,   52,   56;
170,   172,  180,  184,  204,  212,  216,  232,  240;
682,   684,  692,  696,  716,  724,  728,  744,  752,  820,  824, ...
2730, 2732, 2740, 2744, 2764, 2772, 2776, 2792, 2800, 2868, 2872, ...
Triangle T(n,k) in binary:
10;
1010,       1100;
101010,     101100,     110100,     111000;
10101010,   10101100,   10110100,   10111000,   11001100,   11010100, ...
1010101010, 1010101100, 1010110100, 1010111000, 1011001100, 1011010100, ...

Alois P. Heinz, Rows n = 1..11, flattened

First column gives: A020988.
Last elements of rows give: A020522.
Row lengths are: A000081(n+1).
Subsequence of A014486, A031443.
Cf. A061773, A216349, A216350, A216648.

F:= proc(n) option remember; `if`(n=1, [10], sort(map(h->
      parse(cat(1, sort(h)[], 0)), g(n-1, n-1)))) end:
g:= proc(n, i) option remember; `if`(i=1, [[10$n]], [seq(seq(seq(
      [seq (F(i)[w[t]-t+1], t=1..j),v[]], w=combinat[choose](
      [$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])
    end:
b:= proc(n) local h, i, r; h, r:= n/10, 0; for i from 0
      while h>1 do r:= r+2^i*irem(h, 10, 'h') od; r
    end:
T:= proc(n) option remember; map(b, F(n+1))[] end:
seq(T(n), n=1..6);

T(n,k) = A216648(n+1,k)/2 - 2^(2*n).

A258592 Values of k such that the number of rooted trees with k nodes (A000081(k)) is even.

Original entry on oeis.org

0, 3, 4, 6, 7, 9, 11, 12, 13, 19, 20, 21, 24, 26, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 66, 69, 70, 72, 79, 80, 81, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 96, 97, 100, 101, 102, 103, 105, 106, 107

Offset: 1

Vladimir Reshetnikov, Nov 06 2015

nonn

Complement of A263831.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000081, A211192, A263831.

Module[{t}, t[1] = 1; t[k_] := t[k] = Sum[DivisorSum[k-m, t[#] # &] t[m]/(k-1), {m, k-1}]; Select[Range[0, 107], EvenQ@t[#] &]] (* after Alois P. Heinz *)

A263831 Values of k such that the number of rooted trees with k nodes (A000081(k)) is odd.

Original entry on oeis.org

1, 2, 5, 8, 10, 14, 15, 16, 17, 18, 22, 23, 25, 27, 28, 30, 33, 35, 38, 40, 42, 49, 50, 56, 61, 64, 65, 67, 68, 71, 73, 74, 75, 76, 77, 78, 82, 84, 87, 93, 98, 99, 104, 108, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 127, 128, 135, 137, 138, 139

Offset: 1

Vladimir Reshetnikov, Nov 03 2015

nonn

Complement of A258592.

Cf. A000081, A211192, A258592.

Module[{t}, t[1] = 1; t[k_] := t[k] = Sum[DivisorSum[k-m, t[#] # &] t[m]/(k-1), {m, k-1}]; Select[Range[140], OddQ@t[#] &]] (* after Alois P. Heinz *)

A339984 G.f.: g(x) * g(x^2), where g(x) is the g.f. of A000081.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 13, 26, 65, 147, 369, 899, 2298, 5851, 15261, 39945, 105948, 282504, 759480, 2052027, 5576017, 15216998, 41705762, 114715503, 316611401, 876466003, 2433091773, 6771462322, 18889829555, 52809592990, 147935027381, 415182991401, 1167251435240

Offset: 0

Vaclav Kotesovec, Dec 25 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000081, A217781, A309619, A339985.

max = 30; A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[All, 2]]]; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2*k), {k, 1, max}]; CoefficientList[Series[g81 * g81x2, {x, 0, max}], x]

a(n) ~ A339986 * A051491^n / n^(3/2).

A339985 G.f.: g(x)^2 * g(x^2), where g(x) is the g.f. of A000081.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 14, 37, 90, 232, 584, 1512, 3906, 10246, 26984, 71766, 191852, 516400, 1396760, 3797435, 10367628, 28420466, 78183462, 215791426, 597368222, 1658233794, 4614679792, 12872125836, 35982713314, 100787606966, 282832173830, 795070060983

Offset: 0

Vaclav Kotesovec, Dec 25 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000081, A217781, A339984.

max = 30; A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[All, 2]]]; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2*k), {k, 1, max}]; CoefficientList[Series[g81^2 * g81x2, {x, 0, max}], x]

a(n) ~ 2 * A339986 * A051491^n / n^(3/2).

A050395 Reversion of rooted trees A000081.

Original entry on oeis.org

1, -1, 0, 1, -1, 1, -4, 11, -18, 18, 0, -60, 189, -360, 453, -373, 294, -652, 1443, -841, -6127, 27681, -75150, 172371, -389662, 867415, -1630604, 1826409, 1738343, -15855044, 49812000, -109702703, 186277940, -229343877, 86783346, 618570769, -2635628596

Offset: 1

Christian G. Bower, Nov 15 1999

sign

Alois P. Heinz, Table of n, a(n) for n = 1..100
M. Drmota, B. Gittenberger, The shape of unlabeled rooted random trees, Eur. J. Comb. 31 (2010) no 8, 2028-2063, Remark.
N. J. A. Sloane, Transforms
Index entries for reversions of series
Index entries for sequences related to rooted trees

Cf. A050396, A000081.

Module[{t}, t[1] = 1; t[k_] := t[k] = Sum[DivisorSum[k - m, t[#] # &] t[m]/(k-1), {m, k-1}]; InverseSeries[Sum[t[n] x^n, {n, 1, 40}] + O[x]^41][[3]]] (* Vladimir Reshetnikov, Nov 03 2015 *)

A050396 Exponential reversion of rooted trees A000081.

Original entry on oeis.org

1, -1, 1, 1, -14, 64, -97, -1376, 15915, -68820, -467868, 11504693, -85105325, -490922106, 21504158897, -242026388646, -848338391966, 80509815160192, -1351044945506956, 1506218518200935, 472249651021870475, -11783220545826576339, 78093421491085905492

Offset: 1

Christian G. Bower, Nov 15 1999

sign

Alois P. Heinz, Table of n, a(n) for n = 1..100
N. J. A. Sloane, Transforms
Index entries for reversions of series
Index entries for sequences related to rooted trees

Cf. A050395.

a[n:0|1] := n; a[n_] := a[n] = Sum[m a[m] a[n-k*m], {m, n-1}, {k, (n-1)/m}]/(n-1); length = 20; Range[length]! InverseSeries[Sum[a[n] x^n/n!, {n, 0, length}] + O[x]^(length+1)][[3]] (* Vladimir Reshetnikov, Nov 06 2015 *)

A051492 Continued fraction for Otter's rooted tree constant: lim_{n->inf} A000081(n+1)/A000081(n).

Original entry on oeis.org

2, 1, 21, 1, 1, 1, 1, 5, 2, 1, 1, 2, 1, 13, 1, 3, 17, 1, 1, 6, 2, 1796, 1, 1, 2, 110, 41, 2, 3, 1, 7, 43, 11, 7, 15, 3, 3, 3, 2, 4, 1, 4, 2, 1, 35, 1, 1, 1, 2, 6, 1, 1, 1, 1, 4, 3, 2, 1, 17, 7, 1, 5, 2, 7, 1, 8, 3, 12, 1, 6, 2, 1, 1, 5, 2, 4, 3, 1, 17, 7, 30, 3, 1, 4, 1

Offset: 0

N. J. A. Sloane

			2.95576528565199497...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.

S. R. Finch, Otter's Tree Enumeration Constants [Broken link]
S. R. Finch, Otter's Tree Enumeration Constants [Wayback Machine]
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for sequences related to trees
Index entries for sequences related to rooted trees

Cf. A000081, A051491 (decimal expansion).

Offset changed by Andrew Howroyd, Jul 04 2024

A144963 Eigentriangle, row sums = A000081 starting (1, 2, 4, 9, 20, 48, 115, ...).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 3, 2, 2, 4, 9, 8, 3, 4, 4, 9, 20, 16, 8, 6, 8, 9, 20, 48, 41, 16, 16, 12, 18, 20, 48, 115, 98, 41, 32, 32, 27, 40, 115, 286, 250, 98, 82, 64, 72, 60, 96, 115, 286, 719

Offset: 1

Gary W. Adamson, Sep 27 2008

Row sums = A000081 starting with offset 2: (1, 2, 4, 9, 20, 48, 115, ...).
Right border = (1, 1, 2, 4, 9, 20, 48, ...).
Left border = A051573: (1, 1, 1, 2, 3, 8, 16, 41, ...).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle:
   1;
   1,  1;
   1,  1,  2;
   2,  1,  2,  4;
   3,  2,  2,  4,  9;
   8,  3,  4,  4,  9, 20;
  16,  8,  6,  8,  9, 20, 48;
  41, 16, 16, 12, 18, 20, 48, 115;
  98, 41, 32, 32, 27, 40, 48, 115, 286;
  ...
Row 4 = (2, 1, 2, 4) = termwise products of (2, 1, 1, 1) and (1, 1, 2, 4).

Eigentriangle by rows, termwise products of A000081 starting with offset 2: (1, 2, 4, 9, 20, 48, ...) and row terms of an A051573 decrescendo triangle: (1; 1,1; 1,1,1; 2,1,1,1; 3,2,1,1,1; ...) where A051573 = (1, 1, 1, 2, 3, 8, 16, 41, ...).

A185667 Primes in A000081.

Original entry on oeis.org

2, 719, 87811

Offset: 1

Jonathan Vos Post, Feb 08 2011

nonn

A000081(n) is prime for n in A051420.

a(4) = A000081(343) has 158 digits and is too large to be displayed here.

Kevin Ryde, Table of n, a(n) for n = 1..6

Cf. A000040, A000081, A005200, A051420, A119641, A119642.

A000040 INTERSECTION A000081.

a(n) = A000081(A051420(n)). - Amiram Eldar, Nov 11 2017

cp's OEIS Frontend

A216649 Triangle T(n,k) in which n-th row lists in increasing order all positive integers with a representation as totally balanced 2n digit binary string such that all consecutive totally balanced substrings are in nondecreasing order; n>=1, 1<=k<=A000081(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A258592 Values of k such that the number of rooted trees with k nodes (A000081(k)) is even.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A263831 Values of k such that the number of rooted trees with k nodes (A000081(k)) is odd.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A339984 G.f.: g(x) * g(x^2), where g(x) is the g.f. of A000081.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A339985 G.f.: g(x)^2 * g(x^2), where g(x) is the g.f. of A000081.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A050395 Reversion of rooted trees A000081.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A050396 Exponential reversion of rooted trees A000081.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A051492 Continued fraction for Otter's rooted tree constant: lim_{n->inf} A000081(n+1)/A000081(n).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A144963 Eigentriangle, row sums = A000081 starting (1, 2, 4, 9, 20, 48, 115, ...).

Views

Author

Keywords

Comments

Examples

Formula

A185667 Primes in A000081.