A002955 -id:A002955 - cp's OEIS frontend

A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 851, 1928, 4396, 10087, 23273, 53948, 125608, 293543, 688366, 1619087, 3818818, 9029719, 21400706, 50828664, 120963298, 288405081, 688821573, 1647853491, 3948189131, 9473431479

Offset: 1

N. J. A. Sloane

a(n) <= A002955(n). - Max Alekseyev, Sep 23 2009

			From _M. F. Hasler_, Apr 17 2024: (Start)
The table with explicit lists of values starts as follows:
   n | distinct values of 2^...^2 with all possible parenthesizations
-----+---------------------------------------------------------------
   1 | 2
   2 | 2^2 = 4
   3 | (2^2)^2 = 2^(2^2) = 16
   4 | (2^2^2)^2 = 2^8 = 256, (2^2)^(2^2) = 2^(2^2^2) = 2^16 (= 65536)
   5 | 256^2 = 2^16, (2^16)^2 = 2^32, 2^256, 2^2^16 (~ 2*10^19728)
   6 | (2^16)^2 = 2^32, 2^64, 2^512, 2^2^16, 2^2^17, 2^2^32, 2^2^256, 2^2^2^16
   7 | 2^64, 2^128, 2^256, 2^1024, 2^2^17, 2^2^18, 2^2^32, 2^2^33, 2^2^64, 2^2^257,
     | 2^2^512, 2^2^2^16, 2^2^65537, 2^2^2^17, 2^2^2^32, 2^2^2^256, 2^2^2^2^16
  ...| ...
(When parentheses are omitted above, we use that ^ is right associative.) (End)

J. Q. Longyear, personal communication.
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).

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy, with permission)
Sean A. Irvine, Java program (github).
J. Longyear, T. Trotter, and N. J. A. Sloane, Correspondence.
MathOverflow, MathOverflow discussion of related questions.
Vladimir Reshetnikov, Computes terms of OEIS sequence A002845 (C# library).
Jon E. Schoenfield, The 851 values for n=12.
Index entries for sequences related to parenthesizing

Cf. A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

/* Define operators for numbers represented (recursively) as list of positions of bits 1. Illustration using the commands below: T = 3.bits; T.int */
n.bits = vector(hammingweight(n), v,  n -= 1 << v= valuation(n, 2); v.bits)
ONE = 1.bits; m.int = sum(i=1, #m, 1<=0])}
{ADD(m, n, a=#m, b=#n)= if(!a, n, !b, m, a=b=1; until(a>#m|| b>#n, if(m[a]==n[b], until(a>=#m|| m[a]!=m[a+1]|| !#m=m[^a], m[a]=ADD(m[a],ONE)); b++, CMP(m[a], n[b])<0, a++, m=concat([m[1..a-1], [n[b]], m[a..#m]]); b++)); b>#n|| m=concat(m,n[b..#n]); m)}
{CMP(m, n, a=#m, b=#n, c=0)= if(!b, a, !a, -1, while(!(c=CMP(m[a], n[b]))&& a--&& b--, ); if(c, c, 1-b))}
{SUB(m, n, a=#n)= if(!a, m, my(b=a=1, c, i); while(a<=#m && b<=#n, if(0>c=CMP(m[a], n[b]), a++, c, i=[c=n[b]]; b++; while(m[a]!=c=ADD(c, ONE), if(b<=#n && c==n[b], b++, i=concat(i, [c]))); m=concat([m[1..a-1], i, m[a+1..#m]]); a += #i, m=m[^a]; b++)); m)}
A2845 = List([[2.bits]]) /* List of values for each n */
{A002845(n)= while(#A2845= 15. - M. F. Hasler, Apr 28 2024

a(12)-a(13) corrected and a(14)-a(27) added by Jon E. Schoenfield, Oct 11 2008
a(28)-a(29) computed by Kirill Osenkov, added by Vladimir Reshetnikov, Feb 07 2019
a(30)-a(31) added by Sean A. Irvine, Feb 18 2019

A002988 Number of trimmed trees with n nodes.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 2, 3, 6, 10, 21, 39, 82, 167, 360, 766, 1692, 3726, 8370, 18866, 43029, 98581, 227678, 528196, 1232541, 2888142, 6798293, 16061348, 38086682, 90607902, 216230205, 517482053, 1241778985, 2987268628, 7203242490

Offset: 0

N. J. A. Sloane

From Christian G. Bower, Dec 15 1999: (Start)
A trimmed tree is a tree with a forbidden limb of length 2.
A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps. (End)

K. L. McAvaney, personal communication.
A. J. Schwenk, Almost all trees are cospectral, pp. 275-307 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
Index entries for sequences related to trees

Cf. A002955, A002989-A002992, A052318-A052329.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=2, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> `if`(n=0, 1, g(n-1)+(`if`(irem(n, 2, 'r')=0,
         g(r-1), 0)-add(g(i-1)*g(n-i-1), i=1..n-1))/2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2014

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 2, 1, 0]), {d, Divisors[j] }]*g[n-j], {j, 1, n}]/n]; a[n_] := If[n == 0, 1, g[n-1] + (If[Mod[n, 2] == 0, g[Quotient[n, 2]-1], 0] - Sum[g[i-1]*g[n-i-1], {i, 1, n-1}])/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

G.f.: 1 + B(x) + (B(x^2) - B(x)^2)/2 where B(x) is the g.f. of A002955. - Christian G. Bower, Dec 15 1999

a(n) ~ c * d^n / n^(5/2), where d = 2.59952511060090659632378883695..., c = 0.3758284247032014502508501798... . - Vaclav Kotesovec, Aug 24 2014

More terms from Christian G. Bower, Dec 15 1999

A052318 Number of labeled rooted trimmed trees with n nodes.

Original entry on oeis.org

1, 2, 3, 16, 145, 1536, 19579, 290816, 4942305, 94689280, 2020278931, 47523053568, 1222147737265, 34117226135552, 1027550555918475, 33213871550365696, 1146891651823112641, 42135941698113503232, 1641164216596258397347, 67550839668807638712320

Offset: 1

Christian G. Bower, Dec 15 1999

A rooted trimmed tree is a tree with a forbidden limb of length 2.

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..400
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052319-A052329.

A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x,n), polynom) fi end: a:= n-> coeff(A(n+1), x,n)*n!: seq(a(n), n=1..25); # Alois P. Heinz, Aug 23 2008

a[n_] := Sum[ Boole[ EvenQ[n-m]]*(m^((n+m)/2-2)/((n-m)/2)!)*((-1)^((n-m)/2)/(m-1)!), {m, 1, n}]*n!; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Sep 10 2012, after Vladimir Kruchinin *)
Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=sum((if mod(n-m,2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!,m,1,n); /* Vladimir Kruchinin, Aug 07 2012 */

E.g.f. satisfies A(x) = x*exp(A(x) - x^2).
E.g.f.: -LambertW(-x/exp(x^2)). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Jan 08 2014

A052329 Number of rooted trees with a forbidden limb of length 6.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 47, 113, 281, 706, 1807, 4671, 12224, 32247, 85782, 229683, 618767, 1675618, 4559263, 12457483, 34168574, 94040433, 259637564, 718892281, 1995739380, 5553867981, 15490305017, 43293762352, 121235084565

Offset: 1

Christian G. Bower, Dec 15 1999

nonn

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052318-A052328.

Column k=6 of A255636.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=6, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> g(n-1):
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 04 2014

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 6, 1, 0]), {d, Divisors[j]} ]*g[n-j], {j, 1, n}]/n]; a[n_] := g[n-1]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

a(n) satisfies a=SHIFT_RIGHT(EULER(a-b)) where b(6)=1, b(k)=0 if k != 6.

a(n) ~ c * d^n / n^(3/2), where d = 2.95209316333202396584501452688304..., c = 0.43842619727838455589811980703038... . - Vaclav Kotesovec, Aug 25 2014

A002992 Number of n-node trees with a forbidden limb of length 6.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 10, 22, 45, 102, 226, 531, 1253, 3044, 7456, 18604, 46798, 119133, 305567, 790375, 2057523, 5390759, 14200122, 37598572, 100005401, 267131927, 716318650, 1927758155, 5205240762, 14098580633, 38296720823, 104308468102, 284822276099

Offset: 0

N. J. A. Sloane

nonn

A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.

A. J. Schwenk, Almost all trees are cospectral, pp. 275-307 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
Index entries for sequences related to trees

Cf. A002955, A002988-A002991, A052318-A052329.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=6, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> `if`(n=0, 1, g(n-1)+(`if`(irem(n, 2, 'r')=0,
         g(r-1), 0)-add(g(i-1)*g(n-i-1), i=1..n-1))/2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2014

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 6, 1, 0]), {d, Divisors[j] }]*g[n-j], {j, 1, n}]/n]; a[n_] := If[n == 0, 1, g[n-1] + (If[Mod[n, 2 ] == 0, g[Quotient[n, 2]-1], 0] - Sum[g[i-1]*g[n-i-1], {i, 1, n-1}])/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

G.f.: 1 + B(x) + (B(x^2) - B(x)^2)/2 where B(x) is g.f. of A052329.

a(n) ~ c * d^n / n^(5/2), where d = 2.95209316333202396584501452688304..., c = 0.52950413787119576841378912289... . - Vaclav Kotesovec, Aug 25 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999

A255636 Number A(n,k) of n-node rooted trees with a forbidden limb of length k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 3, 4, 0, 1, 1, 2, 4, 7, 8, 0, 1, 1, 2, 4, 8, 15, 17, 0, 1, 1, 2, 4, 9, 18, 35, 36, 0, 1, 1, 2, 4, 9, 19, 43, 81, 79, 0, 1, 1, 2, 4, 9, 20, 46, 102, 195, 175, 0, 1, 1, 2, 4, 9, 20, 47, 110, 251, 473, 395, 0

Offset: 1

Alois P. Heinz, Feb 28 2015

Any rootward k-node path starting at a leaf contains the root or a branching node.

			:    o      o        o      o    o       o      o    o
:  /(|)\    |       / \    /|\   |       |     / \   |
: o ooo 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  o   o    o
: A(6,2) = 8                                        / \
:                                                  o   o
Square array A(n,k) begins:
  1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   2,   2,   2,   2,   2,   2,   2, ...
  0,   2,   3,   4,   4,   4,   4,   4,   4,   4, ...
  0,   4,   7,   8,   9,   9,   9,   9,   9,   9, ...
  0,   8,  15,  18,  19,  20,  20,  20,  20,  20, ...
  0,  17,  35,  43,  46,  47,  48,  48,  48,  48, ...
  0,  36,  81, 102, 110, 113, 114, 115, 115, 115, ...
  0,  79, 195, 251, 273, 281, 284, 285, 286, 286, ...
  0, 175, 473, 625, 684, 706, 714, 717, 718, 719, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-10 give: A063524, A002955, A052321, A052327, A052328, A052329, A255637, A255638, A255639, A255640.
Main diagonal gives A000081.
Cf. A255704.

with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
    end:
A:= (n, k)-> g(n-1, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);

g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[d*(g[d - 1, k] - If[d == k, 1, 0]), {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n]; A[n_, k_] := g[n - 1, k]; Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A255704 Number T(n,k) of n-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 3, 1, 1, 0, 17, 18, 8, 3, 1, 1, 0, 36, 45, 21, 8, 3, 1, 1, 0, 79, 116, 56, 22, 8, 3, 1, 1, 0, 175, 298, 152, 59, 22, 8, 3, 1, 1, 0, 395, 776, 413, 163, 60, 22, 8, 3, 1, 1, 0, 899, 2025, 1131, 450, 166, 60, 22, 8, 3, 1, 1

Offset: 1

Alois P. Heinz, Mar 02 2015

			:    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 o o   o   o
:                       |       |     |   |  / \
:                       o       o     o   o o   o
:                                           |
: T(6,3) = 7                                o
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   1,   1;
  0,   4,   3,   1,  1;
  0,   8,   7,   3,  1,  1;
  0,  17,  18,   8,  3,  1, 1;
  0,  36,  45,  21,  8,  3, 1, 1;
  0,  79, 116,  56, 22,  8, 3, 1, 1;
  0, 175, 298, 152, 59, 22, 8, 3, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=1-10 give: A063524, A002955 (for n>1), A318899, A318900, A318901, A318902, A318903, A318904, A318905, A318906.
Row sums give A000081.
T(2*n+1,n+1) gives A255705.
Cf. A255636.

with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> g(n-1, k) -`if`(k=1, 0, g(n-1, k-1)):
seq(seq(T(n, k), k=1..n), n=1..14);

g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[#-1, k] - If[# == k, 1, 0])&] * g[n-j, k], {j, 1, n}]/n];
T[n_, k_] := g[n-1, k] - If[k == 1, 0, g[n-1, k-1]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

T(n,1) = A255636(n,1), T(n,k) = A255636(n,k) - A255636(n,k-1) for k>1.

A052319 Number of increasing rooted trimmed trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 7, 28, 131, 720, 4513, 31824, 249513, 2151744, 20242983, 206313024, 2264425179, 26628836352, 334022337153, 4451717814528, 62820790592913, 935750983412736, 14672143677452679, 241555066200437760

Offset: 1

Christian G. Bower, Dec 11 1999

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
A trimmed tree is a tree with a forbidden limb of length 2.
A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.
Number of permutations on [n+1] beginning with 12 and avoiding a consecutive 132 pattern (n>=1). For example, a(4)=2 counts 12345, 12453. - Ralf Stephan, Apr 25 2004

Vaclav Kotesovec, Table of n, a(n) for n = 1..465
M. E. Jones and J. B. Remmel, Pattern matching in the cycle structures of permutations, Pure Math. Appl. (PU.M.A.), 22 (2011) 173-208.
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021, p. 7.
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052318-A052329.

seq(n! * coeff(series(-log(1-sqrt(Pi/2)*erf(x/sqrt(2))), x, n+1), x, n), n=1..20) # Vaclav Kotesovec, Jan 07 2014

Rest[CoefficientList[Series[-Log[1-Sqrt[Pi/2]*Erf[x/Sqrt[2]]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 07 2014 *)

E.g.f.: A(x) = 1/B(-x) where B'(x) is e.g.f. of A006882 and B(0) = 1.
E.g.f.: A(x) satisfies A'(x) = exp(A(x)-x^2/2).
E.g.f.: exp(-x^2/2)/(1-int[0..x, exp(-x^2/2)]). - Ralf Stephan, Apr 25 2004
E.g.f.: -log(1-sqrt(Pi/2)*erf(x/sqrt(2))). - Vaclav Kotesovec, Jan 07 2014
Limit n->infinity (a(n)/n!)^(1/n) = 1/(sqrt(2)*InverseErf(sqrt(2/Pi))) = 1/A240885 = 0.7839769312035474991... - Vaclav Kotesovec, Jan 07 2014
a(n) ~ (n-1)! / (sqrt(2)*InverseErf(sqrt(2/Pi)))^n. - Vaclav Kotesovec, Aug 22 2014

Formula updated by Christian G. Bower, Mar 06 2001

A052321 Number of rooted trees with a forbidden limb of length 3.

Original entry on oeis.org

1, 1, 2, 3, 7, 15, 35, 81, 195, 473, 1171, 2924, 7396, 18848, 48446, 125311, 326145, 853188, 2242616, 5919197, 15683008, 41694334, 111195166, 297393668, 797475499, 2143631474, 5775002574, 15590201095, 42168292074, 114260967888, 310124721255, 843053354234

Offset: 1

Christian G. Bower, Dec 15 1999

nonn

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.
Likely a duplicate of A003006. - R. J. Mathar, Mar 23 2012
Only first 10 terms match, but then a(11) = 1171, and A003006(11) = 1170. - Vladimir Reshetnikov, Mar 05 2019

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A003006 (first 10 terms match), A052318-A052329.

Column k=3 of A255636.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=3, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> g(n-1):
seq(a(n), n=1..35);  # Alois P. Heinz, Jun 26 2014

g[n_] := g[n] = If[n==0, 1, Sum[DivisorSum[j, #*(g[#-1] - If[#==3, 1, 0])&] * g[n-j], {j, 1, n}]/n];
a[n_] := g[n-1];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Apr 04 2017, after Alois P. Heinz *)

a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(3)=1, b(k)=0 if k != 3.

a(n) ~ c * d^n / n^(3/2), where d = 2.851157026715821487965080545784048..., c = 0.4192933669718878505916053142459... . - Vaclav Kotesovec, Aug 24 2014

A052328 Number of rooted trees with a forbidden limb of length 5.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 46, 110, 273, 684, 1747, 4505, 11763, 30956, 82153, 219437, 589747, 1593170, 4324445, 11787195, 32251520, 88548011, 243877256, 673605521, 1865445693, 5178574184, 14408195935, 40170674295, 112213616851

Offset: 1

Christian G. Bower, Dec 15 1999

nonn

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052318-A052329.

Column k=5 of A255636.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=5, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> g(n-1):
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 04 2014

g[n_] := g[n] = If[n==0, 1, Sum[Sum[d(g[d-1] - If[d==5, 1, 0]), {d, Divisors[j]}] g[n-j], {j, 1, n}]/n];
a[n_] := g[n-1];
Array[a, 35] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(5)=1, b(k)=0 if k != 5.

a(n) ~ c * d^n / n^(3/2), where d = 2.944791657501974377513779510930324..., c = 0.43624554592719796037836168844839... . - Vaclav Kotesovec, Aug 25 2014

cp's OEIS Frontend

A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).

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A002988 Number of trimmed trees with n nodes.

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A052318 Number of labeled rooted trimmed trees with n nodes.

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Maxima

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A052329 Number of rooted trees with a forbidden limb of length 6.

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A002992 Number of n-node trees with a forbidden limb of length 6.

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A255636 Number A(n,k) of n-node rooted trees with a forbidden limb of length k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A255704 Number T(n,k) of n-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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