A032200 -id:A032200 - cp's OEIS frontend

A118376 Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node.

1, 2, 6, 24, 112, 568, 3032, 16768, 95200, 551616, 3248704, 19389824, 117021824, 712934784, 4378663296, 27081760768, 168530142720, 1054464293888, 6629484729344, 41860283723776, 265346078982144, 1687918305128448, 10771600724946944, 68941213290561536

Offset: 1

Jeremy Johnson (jjohnson(AT)cs.drexel.edu), May 15 2006

nonn

The number of trees with leaf nodes equal to 1 is counted by the sequence A001003 of super-Catalan numbers. The number of binary trees is counted by the sequence A007317 and the number of binary trees with leaf nodes equal to 1 is counted by the sequence A000108 of Catalan numbers.
Also the number of series-reduced enriched plane trees of weight n. A series-reduced enriched plane tree of weight n is either the number n itself or a finite sequence of at least two series-reduced enriched plane trees, one of each part of an integer composition of n. For example, the a(3) = 6 trees are: 3, (21), (12), (111), ((11)1), (1(11)). - Gus Wiseman, Sep 11 2018
Conjectured to be the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 3>4, 3>5} of length 5. That is, conjectured to be the number of length n permutations having no subsequences of length 5 in which the first element is the largest, and the third element is larger than the fourth and fifth elements. - Sergey Kitaev, Dec 13 2020
This conjecture has been proven. It can be restated as the number of size n permutations avoiding 51423, 51432, 52413, 52431, 53412, 53421, 54312, 54321. There are twelve sets of permutations avoiding eight size five permutations that are known to match this sequence. A further four are conjectured to match this sequence. - Christian Bean, Jul 24 2024

			T(3) = 6 because there are six trees
  3    3      3     3     3       3
      2 1    2 1   1 2   1 2    1 1 1
            1 1           1 1
From _Gus Wiseman_, Sep 11 2018: (Start)
The a(4) = 24 series-reduced enriched plane trees:
  4,
  (31), (13), (22), (211), (121), (112), (1111),
  ((21)1), ((12)1), (1(21)), (1(12)), (2(11)), ((11)2),
  ((111)1), (1(111)), ((11)(11)), ((11)11), (1(11)1), (11(11)),
  (((11)1)1), ((1(11))1), (1((11)1)), (1(1(11))).
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, PermPAL database.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Hadrien Cambazard and Nicolas Catusse, Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane, arXiv preprint arXiv:1512.06649 [cs.DS], 2015.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Pawel Hitczenko, Jeremy R. Johnson, and Hung-Jen Huang, Distribution of a class of divide and conquer recurrences arising from the computation of the Walsh-Hadamard transform, Theoretical Computer Science, Vol. 352, 2006, pp. 8-30.
J. R. Johnson and M. Püschel, In search for the optimal Walsh-Hadamard transform, Proc. ICASSP, Vol. 4, 2000, pp. 3347-3350.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A000108, A000311, A001003, A005804, A007317, A007853, A032171, A032200, A143363, A289501, A317852.

T := proc(n) option remember; local C, s, p, tp, k, i; if n = 1 then return 1; else s := 1; for k from 2 to n do C := combinat[composition](n,k); for p in C do tp := map(T,p); s := s + mul(tp[i],i=1..nops(tp)); end do; end do; end if; return s; end;

Rest[CoefficientList[Series[(Sqrt[1-8*x+8*x^2]-1)/(4*x-4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 03 2014 *)
a[n_] := 1+Sum[Binomial[n-1, k-1]*Hypergeometric2F1[2-k, k+1, 2, -1], {k, 2, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
urp[n_]:=Prepend[Join@@Table[Tuples[urp/@ptn],{ptn,Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[urp[n]],{n,7}] (* Gus Wiseman, Sep 11 2018 *)

a(n):=sum((-1)^j*2^(n-j-1)*binomial(n,j)*binomial(2*n-2*j-2,n-2*j-1),j,0,(n-1)/2)/n; /* Vladimir Kruchinin, Sep 29 2020 */

x='x+O('x^25); Vec((sqrt(1-8*x+8*x^2) - 1)/(4*x-4)) \\ G. C. Greubel, Feb 08 2017

Recurrence: T(1) = 1; For n > 1, T(n) = 1 + Sum_{n=n1+...+nt} T(n1)*...*T(nt).
G.f.: (-1+(1-8*z+8*z^2)^(1/2))/(-4+4*z).
From Vladimir Kruchinin, Sep 03 2010: (Start)
O.g.f.: A(x) = A001003(x/(1-x)).
a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A001003(k), n>0. (End)
D-finite with recurrence: n*a(n) + 3*(-3*n+4)*a(n-1) + 4*(4*n-9)*a(n-2) + 8*(-n+3)*a(n-3) = 0. - R. J. Mathar, Sep 27 2013
a(n) ~ sqrt(sqrt(2)-1) * 2^(n-1/2) * (2+sqrt(2))^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 03 2014
From Peter Bala, Jun 17 2015: (Start)
With offset 0, binomial transform of A001003.
O.g.f. A(x) = series reversion of x*(2*x - 1)/(2*x^2 - 1); 2*(1 - x)*A^2(x) - A(x) + x = 0.
A(x) satisfies the differential equation (x - 9*x^2 + 16*x^3 - 8*x^4)*A'(x) + x*(3 - 4*x)*A(x) + x*(2*x - 1) = 0. Extracting coefficients gives Mathar's recurrence above. (End)
a(n) = Sum_{j=0..(n-1)/2} (-1)^j*2^(n-j-1)*C(n,j)*C(2*n-2*j-2,n-2*j-1)/n. - Vladimir Kruchinin, Sep 29 2020

A055340 Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 8, 4, 1, 0, 1, 9, 19, 16, 5, 1, 0, 1, 12, 37, 46, 25, 6, 1, 0, 1, 16, 66, 118, 96, 40, 7, 1, 0, 1, 20, 110, 260, 300, 184, 56, 8, 1, 0, 1, 25, 172, 527, 811, 688, 318, 80, 9, 1, 0, 1, 30, 257, 985, 1951, 2178, 1408, 524, 105, 10, 1, 0

Offset: 1

Christian G. Bower, May 14 2000

			G.f. = x^(y + x*y + x^2*(y + y^2) + x^3*(y + 2*y^2 + y^3) + x^4*(y + 4*y^2 + 3*y^3 + y^4) + ...).
n\k 1  2  3  4  5  6  7  8
--:-- -- -- -- -- -- -- --
1:  1
2:  1  0
3:  1  1  0
4:  1  2  1  0
5:  1  4  3  1  0
6:  1  6  8  4  1  0
7:  1  9 19 16  5  1  0
8:  1 12 37 46 25  6  1  0

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Row sums give A032200.
Columns 2..8 are A002620(n-1), A055341, A055342, A055343, A055344, A055345, A055346.
Cf. A055347, A055348, A055349, A055356, A055363.

m = 13; A[, ] = 0;
Do[A[x_, y_] = x (y - Sum[EulerPhi[i]/i Log[1 - A[x^i, y^i]], {i, 1, m}]) + O[x]^m + O[y]^m // Normal, {m}];
Join[{1}, Append[CoefficientList[#/y, y], 0]& /@ Rest @ CoefficientList[ A[x, y]/x, x]] // Flatten (* Jean-François Alcover, Oct 02 2019 *)

{T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*y - x*sum(i=1, j, eulerphi(i)/i * log(1 - subst( subst( A + x * O(x^min(j, n\i)), x, x^i), y, y^i) ) )); polcoeff( polcoeff(A, n), k))}; /* Michael Somos, Aug 24 2015 */

G.f. satisfies A(x, y)=xy+x*CIK(A(x, y))-x. Shifts up under CIK transform.
G.f. satisfies A(x, y) = x*(y - Sum_{i>0} phi(i)/i * log(1 - A(x^i, y^i))). - Michael Somos, Aug 24 2015
Sum_k T(n, k) = A032200(n). - Michael Somos, Aug 24 2015

A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.

Original entry on oeis.org

1, 2, 9, 68, 730, 10164, 173838, 3524688, 82627200, 2198295360, 65431163160, 2154106470240, 77714083773456, 3048821300491680, 129221979665461200, 5884296038166954240, 286492923374605966080, 14851359950834255500800

Offset: 1

Christian G. Bower, Sep 15 1998

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.83).

Alois P. Heinz, Table of n, a(n) for n = 1..140
C. G. Bower, Transforms (2)
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 454
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Index entries for sequences related to mobiles

Cf. A029768, A032200, A055349.

logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: b:= logtr(-a): a:= n-> `if`(n<=1,1, -n*b(n-1)): seq(a(n), n=1..25); # Alois P. Heinz, Sep 14 2008

a[n_] = Sum[Binomial[n, j]*Abs[StirlingS1[n-1, j]]*j!, {j, 0, n}]; Array[a, 18]
(* Jean-François Alcover, Jun 22 2011, after Vladimir Kruchinin *)

Vec(serlaplace(serreverse(x/(1 - log(1-x + O(x^20)))))) \\ Andrew Howroyd, Sep 19 2018

Divides by n and shifts left under "CIJ" (necklace, indistinct, labeled) transform.
E.g.f. A(x) satisfies A(x) = x-x*log(1-A(x)). [Corrected by Andrey Zabolotskiy, Sep 16 2022]
a(n) = Sum_{j=0..n} binomial(n,j)*abs(Stirling1(n-1,j))*j!, n > 0. - Vladimir Kruchinin, Feb 03 2011
a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^(n-1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013
E.g.f.: series reversion of x/(1 - log(1-x)). - Andrew Howroyd, Sep 19 2018

A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 23, 59, 148, 385, 1006, 2678, 7170, 19421, 52933, 145364, 401421, 1114713, 3109710, 8713076, 24506121, 69168705, 195849114, 556165311, 1583601840, 4520226558, 12931917204, 37075154703

Offset: 1

Christian G. Bower

Also the number of locally Lyndon plane trees with n nodes, where a plane tree is locally Lyndon if the sequence of branches directly under any given node is a Lyndon word. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 10 locally Lyndon plane trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  ((oo(o)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Wikipedia, Lyndon word
Index entries for sequences related to mobiles

Cf. A032200, A055363.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

T[n_, k_] := Module[{A}, A[, ] = 0; If[k < 1 || k > n, 0, For[j = 1, j <= n, j++, A[x_, y_] = x*y - x*Sum[MoebiusMu[i]/i * Log[1 -  A [x^i, y^i]] + O[x]^j // Normal , {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
a[n_] := a[n] = Sum[T[n, k], {k, 1, n}];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 28}] (* Jean-François Alcover, Jun 30 2017, using Michael Somos' code for A055363 *)
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
lynplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lynplane/@c],LyndonQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lynplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CHK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CHK" (necklace, identity, unlabeled) transform.
From Petros Hadjicostas, Dec 03 2017: (Start)
a(n+1) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = c(1) = 1.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+1)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).
The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 147*x^6 + 414*x^7 + 1203*x^8 + ...
(End)

A108531 Number of mobiles (cycle rooted trees) with n nodes and 2-colored internal (non-leaf) nodes.

Original entry on oeis.org

1, 2, 6, 18, 60, 206, 770, 2950, 11748, 47746, 197808, 830878, 3532790, 15168294, 65683552, 286504378, 1257693038, 5551978426, 24630911086, 109759215338, 491060888588, 2204938828766, 9933016712348, 44881199711338

Offset: 1

Christian G. Bower, Jun 07 2005

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for sequences related to mobiles
C. G. Bower, Transforms (2)

Cf. A032200, A108530, A108532.

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+2*CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left and halves under CIK transform.

A317852 Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.

Original entry on oeis.org

1, 1, 1, 3, 8, 26, 76, 247, 783, 2565, 8447, 28256, 95168, 323720, 1108415, 3821144, 13246307, 46158480, 161574043, 567925140, 2003653016, 7092953340, 25186731980, 89690452750, 320221033370, 1146028762599, 4110596336036, 14774346783745, 53203889807764, 191934931634880

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

Also the number of plane trees with n nodes where the sequence of branches directly under any given node has relatively prime run-lengths.

			The a(5) = 8 locally aperiodic plane trees:
  ((((o)))),
  (((o)o)), ((o(o))), (((o))o), (o((o))),
  ((o)oo), (o(o)o), (oo(o)).
The a(6) = 26 locally aperiodic plane trees:
  (((((o)))))  ((((o)o)))  (((o)oo))  ((o)ooo)
               (((o(o))))  ((o(o)o))  (o(o)oo)
               ((((o))o))  ((oo(o)))  (oo(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))oo)
                           (o((o))o)
                           (oo((o)))
                           ((o)(o)o)
                           ((o)o(o))
                           (o(o)(o))

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

Cf. A000108, A000837, A007853, A032171, A032200, A254040, A301700, A303386, A303431, A304173, A304175, A317708, A317852.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
aperplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[aperplane/@c],aperQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[aperplane[n]],{n,10}]

Tfm(p, n)={sum(d=1, n, moebius(d)*(subst(1/(1+O(x*x^(n\d))-p), x, x^d)-1))}
seq(n)={my(p=O(1)); for(i=1, n, p=1+Tfm(x*p, i)); Vec(p)} \\ Andrew Howroyd, Feb 08 2020

a(16)-a(17) from Robert Price, Sep 15 2018

Terms a(18) and beyond from Andrew Howroyd, Feb 08 2020

A108526 Number of mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 2, 5, 16, 54, 210, 841, 3555, 15402, 68336, 308206, 1410175, 6525500, 30492934, 143669529, 681781043, 3255653089, 15632422715, 75429279214, 365556955492, 1778608580060, 8684658137204, 42543288504844, 209022441144144

Offset: 1

Christian G. Bower, Jun 07 2005

nonn

A generator is a leaf or a node with just one child.

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A108521-A108529, A032200, A032203.

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=x); for(n=2, n, p += x^n*polcoef(x*p + CIK(p, n), n)); Vecrev(p/x)} \\ Andrew Howroyd, Aug 31 2018

G.f. satisfies (2-x)*A(x) = x - 1 + CIK(A(x)).

A032202 Sequence (a(n): n >= 1) that shifts left 2 places under the "CIK" (necklace, indistinct, unlabeled) transform and satisfies a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 22, 41, 92, 193, 435, 963, 2215, 5051, 11754, 27375, 64381, 151898, 360661, 859149, 2055804, 4934428, 11883930, 28699336, 69497354, 168691424, 410399073, 1000486306, 2443761830, 5979742904, 14656709518

Offset: 1

Christian G. Bower

From Petros Hadjicostas, Dec 30 2018: (Start)
a(n+2) = (1/n)*Sum_{d|n} phi(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = a(2) = 1, c(1) = 1, and c(2) = 3.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+2)*x^n = -Sum_{n>=1} (phi(n)/n)*log(1-A(x^n)).
The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 46*x^5 + 117*x^6 + 281*x^7 + 707*x^8 + 1717*x^9 + 4288*x^10 + 10583*x^11 + 26401*x^12 + ...
(End)
The first two terms of the sequence must be specified. In general, if the sequence (b(n): n >= 1) is such that (b(n+2): n >= 1) = CIK((b(n): n >= 1)), then b(3) = b(1), b(4) = (1/2)*(b(1)^2 + 2*b(2) + b(1)), b(5) = (b(1)/3)*(b(1)^2 + 3*b(2) + 5), and so on. - Petros Hadjicostas, Jan 01 2019

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A032200, A032201, A108531.

m = 33; a[1] = a[2] = 1; A[_] = 0;
Do[A[x_] = x(a[1] + x a[2] - x Sum[EulerPhi[n] Log[1-A[x^n]]/n, {n, 1, m}]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Sep 17 2019 *)

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=1+O(x));for(i=1, n\2, p=1+x+x*CIK(x*p, 2*i)); Vec(p+O(x^n))} \\ Andrew Howroyd, Jun 20 2018

Name modified by Petros Hadjicostas, Jan 01 2019

A106364 Mobiles (cycle rooted trees) where no branch is identical to its adjacent neighbor.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 14, 30, 68, 159, 381, 914, 2238, 5508, 13701, 34288, 86401, 218818, 557067, 1424083, 3655221, 9414642, 24328133, 63049458, 163844470, 426831429, 1114496370, 2916228670, 7645777113, 20082543578, 52839735409, 139251228967

Offset: 1

Christian G. Bower, Apr 29 2005

Index entries for sequences related to mobiles

Cf. A032200.

Shifts left under CycleBG transform.
CycleBG transform T(A) = invMOEBIUS(invEULER(Carlitz(A)) + A(x^2) - A) + A.
Carlitz transform T(A(x)) has g.f. 1/(1-sum(k>0, (-1)^(k+1)*A(x^k))).
General formula for the CycleBG transform: T(A)(x) = A(x) - Sum_{k>=0} A(x^{2k+1}) + Sum_{k>=1} (phi(k)/k)*log(Carlitz(A)(x^k)). For a proof, see the links in the documentation of sequence A106368. - Petros Hadjicostas, Oct 08 2017

A124345 Number of mobiles (circular rooted trees) on n nodes with thinning limbs.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 24, 47, 113, 258, 624, 1492, 3694, 9090, 22753, 57111, 144541, 367243, 938449, 2406720, 6198045, 16013447, 41507254, 107887092, 281170859, 734518306, 1923119062, 5045423169, 13262334340, 34923020733, 92113656841

Offset: 1

Christian G. Bower, Oct 30 2006, suggested by Franklin T. Adams-Watters

nonn

A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Cf. A032200, A124343-A124348.

cp's OEIS Frontend

A118376 Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node.

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A055340 Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

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A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.

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A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

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A108531 Number of mobiles (cycle rooted trees) with n nodes and 2-colored internal (non-leaf) nodes.

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A317852 Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.

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A108526 Number of mobiles (cycle rooted trees) with n generators.

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