A002992 -id:A002992 - cp's OEIS frontend

A002988 Number of trimmed trees with n nodes.

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

A002955 Number of (unordered, unlabeled) rooted trimmed trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 79, 175, 395, 899, 2074, 4818, 11291, 26626, 63184, 150691, 361141, 869057, 2099386, 5088769, 12373721, 30173307, 73771453, 180800699, 444101658, 1093104961, 2695730992, 6659914175, 16481146479, 40849449618

Offset: 1

N. J. A. Sloane

A rooted trimmed tree is a tree without limbs of length >= 2. Limbs are the paths from the leafs (towards the root) to the nearest branching point (with the root considered to be a branching point). [clarified by Joerg Arndt, Mar 03 2015]
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.
Also counts the unordered rooted trees without "x x" in the level sequence for the pre-order walk. The bijection transforms the two outmost nodes in all limbs of lengths >= 2 into V-shaped subtrees. - Joerg Arndt, Mar 03 2015

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 = 1..1000 (terms n = 1..300 from Vincenzo Librandi)
F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
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)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002988-A002992, A052318-A052329.

Column k=2 of A255636.

with(numtheory): a:= proc(n) option remember; local d,j,aa; aa:= n-> a(n)-`if`(n=2,1,0); if n<=1 then n else (add(d*aa(d), d=divisors(n-1)) +add(add(d*aa(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..32); # Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = (Total[ #*b[#]& /@ Divisors[n-1] ] + Sum[ Total[ #*b[#]& /@ Divisors[j] ]*a[n-j], {j, 1, n-2}]) / (n-1); a[1] = 1; b[n_] := a[n]; b[2] = 0; Table[ a[n], {n, 1, 32}](* Jean-François Alcover, Nov 18 2011, after Maple *)

a(n) satisfies a=SHIFT_RIGHT(EULER(a-b)) where b(2)=1, b(k)=0 if k != 2.

a(n) ~ c * d^n / n^(3/2), where d = 2.59952511060090659632378883695107..., c = 0.391083882871301267612387143401... . - Vaclav Kotesovec, Aug 24 2014

More terms, formula and comments 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

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

A052322 Number of labeled rooted trees with a forbidden limb of length 3.

Original entry on oeis.org

1, 2, 9, 40, 385, 4536, 66409, 1127792, 21981537, 483858640, 11873508361, 321497975448, 9522483900241, 306292854886760, 10632656242583145, 396223803663328096, 15776491521834720961, 668460175137505993248, 30030668624358362706697, 1425868954034374729854920

Offset: 1

Christian G. Bower, Dec 15 1999

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..390
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

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

Rest[CoefficientList[Series[-LambertW[-x*E^(-x^3)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

E.g.f. satisfies A(x) = x*exp(A(x) - x^3).
E.g.f: -LambertW(-x*exp(-x^3)). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ sqrt(1+LambertW(-3*exp(-3))) * n^(n-1) * exp(n/3*LambertW(-3*exp(-3))). - Vaclav Kotesovec, Jan 08 2014

A002989 Number of n-node trees with a forbidden limb of length 3.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 7, 14, 28, 61, 131, 297, 678, 1592, 3770, 9096, 22121, 54451, 135021, 337651, 849698, 2152048, 5479408, 14022947, 36048514, 93061268, 241160180, 627179689, 1636448181, 4282964600, 11241488853, 29584389474

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-A002992, A052318-A052329.

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-> `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 == 3, 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 A052321.

a(n) ~ c * d^n / n^(5/2), where d = 2.851157026715821487965080545784..., c = 0.463162985533004672966744142107... . - Vaclav Kotesovec, Aug 24 2014

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

A002991 Number of n-node trees with a forbidden limb of length 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 10, 21, 43, 97, 215, 503, 1187, 2876, 7033, 17510, 43961, 111664, 285809, 737632, 1915993, 5008652, 13163785, 34774873, 92282214, 245930746, 657931603, 1766481135, 4758553683, 12858286083, 34844908142, 94681272368

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. - Christian G. Bower, Dec 15 1999

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-A002992, A052318-A052329.

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-> `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 == 5, 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 A052328. - Christian G. Bower, Dec 15 1999

a(n) ~ c * d^n / n^(5/2), where d = 2.9447916575019743775137795109303..., c = 0.521642401804532770865780146005... . - Vaclav Kotesovec, Aug 25 2014

More terms from Christian G. Bower, Dec 15 1999

cp's OEIS Frontend

A002988 Number of trimmed trees with n nodes.

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A002955 Number of (unordered, unlabeled) rooted trimmed trees with n nodes.

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A052318 Number of labeled rooted trimmed trees with n nodes.

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

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A052319 Number of increasing rooted trimmed trees with n nodes.

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

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Maple

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

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