A320154 -id:A320154 - cp's OEIS frontend

A079500 Number of compositions of the integer n in which the first part is >= the other parts.

1, 1, 2, 3, 5, 8, 14, 24, 43, 77, 140, 256, 472, 874, 1628, 3045, 5719, 10780, 20388, 38674, 73562, 140268, 268066, 513350, 984911, 1892875, 3643570, 7023562, 13557020, 26200182, 50691978, 98182666, 190353370, 369393466, 717457656, 1394632365, 2713061899

Offset: 0

Arnold Knopfmacher, Jan 21 2003

nonn

Essentially the same as A007059: a(n) = A007059(n+1).
In lunar arithmetic in base 2, this is the number of lunar divisors of the number 111...1 (with n 1's). E.g., 1111 has a(4) = 5 divisors (see A048888). - N. J. A. Sloane, Feb 23 2011.
First differences of A186537. - N. J. A. Sloane, Feb 23 2011
Number of balanced ordered rooted trees with n non-root nodes (see A048816 for unordered balanced trees); see example. The compositions are obtained from the level sequences by identifying a length-k run of (non-root) levels [t, t+1, t+2, ..., t+k-1] with a part k. - Joerg Arndt, Jul 20 2014

			From _Joerg Arndt_, Dec 29 2012: (Start)
There are a(7)=24 compositions p(1)+p(2)+...+p(m)=7 such that p(k) <= p(1):
[ 1]  [ 1 1 1 1 1 1 1 ]
[ 2]  [ 2 1 1 1 1 1 ]
[ 3]  [ 2 1 1 1 2 ]
[ 4]  [ 2 1 1 2 1 ]
[ 5]  [ 2 1 2 1 1 ]
[ 6]  [ 2 1 2 2 ]
[ 7]  [ 2 2 1 1 1 ]
[ 8]  [ 2 2 1 2 ]
[ 9]  [ 2 2 2 1 ]
[10]  [ 3 1 1 1 1 ]
[11]  [ 3 1 1 2 ]
[12]  [ 3 1 2 1 ]
[13]  [ 3 1 3 ]
[14]  [ 3 2 1 1 ]
[15]  [ 3 2 2 ]
[16]  [ 3 3 1 ]
[17]  [ 4 1 1 1 ]
[18]  [ 4 1 2 ]
[19]  [ 4 2 1 ]
[20]  [ 4 3 ]
[21]  [ 5 1 1 ]
[22]  [ 5 2 ]
[23]  [ 6 1 ]
[24]  [ 7 ]
(End)
From _Joerg Arndt_, Jul 20 2014: (Start)
The a(7) = 24 balanced ordered rooted trees with 7 non-root nodes are, as level sequences (of the pre-order walk):
01:  [ 0 1 1 1 1 1 1 1 ]
02:  [ 0 1 2 1 2 1 2 2 ]
03:  [ 0 1 2 1 2 2 1 2 ]
04:  [ 0 1 2 1 2 2 2 2 ]
05:  [ 0 1 2 2 1 2 1 2 ]
06:  [ 0 1 2 2 1 2 2 2 ]
07:  [ 0 1 2 2 2 1 2 2 ]
08:  [ 0 1 2 2 2 2 1 2 ]
09:  [ 0 1 2 2 2 2 2 2 ]
10:  [ 0 1 2 3 1 2 3 3 ]
11:  [ 0 1 2 3 2 3 2 3 ]
12:  [ 0 1 2 3 2 3 3 3 ]
13:  [ 0 1 2 3 3 1 2 3 ]
14:  [ 0 1 2 3 3 2 3 3 ]
15:  [ 0 1 2 3 3 3 2 3 ]
16:  [ 0 1 2 3 3 3 3 3 ]
17:  [ 0 1 2 3 4 2 3 4 ]
18:  [ 0 1 2 3 4 3 4 4 ]
19:  [ 0 1 2 3 4 4 3 4 ]
20:  [ 0 1 2 3 4 4 4 4 ]
21:  [ 0 1 2 3 4 5 4 5 ]
22:  [ 0 1 2 3 4 5 5 5 ]
23:  [ 0 1 2 3 4 5 6 6 ]
24:  [ 0 1 2 3 4 5 6 7 ]
(End)
From _Gus Wiseman_, Oct 07 2018: (Start)
The a(0) = 1 through a(6) = 14 balanced rooted plane trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((oo)(o))    ((oo)(oo))
                                     ((((oo))))   ((ooo)(o))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o)(oo)))
                                                  (((oo)(o)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))
(End)

Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295.

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 400 terms from T. D. Noe)
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006).
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.1.
R. Kemp, Balanced ordered trees, Random Structures Algorithms, 5 (1994), pp. 99-121.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A188541.
Cf. A000081 A000311, A001678, A120803, A320154, A320160, A316624, A320169.
Cf. A092921.

M:=101:
t1:=add( (1-x)*x^k/(1-2*x+x^k), k=1..M):
series(t1,x,M-1);
seriestolist(%);
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1,
      `if`(m=0, add(b(n-j, j), j=1..n),
      add(b(n-j, min(n-j, m)), j=1..min(n, m))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, May 01 2014

nn=36;CoefficientList[Series[Sum[x^i/(1-(x-x^(i+1))/(1-x)),{i,0,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 12 2013 *)
b[n_, m_] := b[n, m] = If[n==0, 1, If[m==0, Sum[b[n-j, j], {j, 1, n}], Sum[ b[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

G.f.: (1-z) * Sum_{k>=0} z^k/(1 - 2*z + z^(k+1)).
a(n) = A048888(n) - 1.
This is a subsequence of A067399: a(n) = A067399(2^n-1).
G.f.: -((1 + x^2 + 1/(x-1))/x)*( 1 + x*(x-1)^3*(1-x+x^3)/( Q(0) - x*(x-1)^3*(1-x+x^3)) ), where Q(k) = (x+1)*(2*x-1)*(1-x)^2 + x^(k+2)*(x+x^2+x^3-2*x^4-1 - x^(k+3) + x^(k+5)) - x*(-1+2*x-x^(k+3))*(1-2*x+x^2+x^(k+4)-x^(k+5))*(-1+4*x-5*x^2+2*x^3 - x^(k+2)- x^(k+5) + 2*x^(k+3) - x^(2*k+5) + x^(2*k+6))/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 14 2013
a(n) = Sum_{j=1..n} F(j, n+1-j), where F(n,k) is the n-th k-generalized Fibonacci number A092921(k,n). - Gregory L. Simay, Aug 21 2022

Offset corrected by N. J. A. Sloane, Feb 23 2011
More terms from N. J. A. Sloane, Feb 24 2011
Further edits (required in order to clarify the definition - is the first part >= the rest. or only > the rest? Answer: the former; for the latter, see A007059) by N. J. A. Sloane, May 08 2011

A048816 Number of rooted trees with n nodes with every leaf at the same height.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 28, 42, 68, 103, 168, 260, 420, 665, 1075, 1716, 2787, 4489, 7304, 11849, 19333, 31504, 51561, 84347, 138378, 227096, 373445, 614441, 1012583, 1669774, 2756951, 4555183, 7533988, 12469301, 20655523, 34238310, 56795325, 94270949

Offset: 1

Christian G. Bower, Apr 15 1999

nonn

The trees are unordered (see A000081). For balanced ordered rooted trees see A079500. - Joerg Arndt, Jul 20 2014

The trees are unlabeled. For labeled version see A238372. - Alois P. Heinz, Dec 29 2014

			See Arndt link.
From _Gus Wiseman_, Oct 08 2018: (Start)
The a(1) = 1 through a(7) = 12 balanced rooted trees with n nodes:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((((oo))))   ((oo)(oo))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o)(oo)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..3500 (terms 1..300 from Alois P. Heinz)
Joerg Arndt, balanced unordered rooted trees for n = 1..10
Index entries for sequences related to rooted trees

Cf. A048808-A048815.
Row sums of A244925.
Cf. A079500, A238372.
Cf. A000081, A000311, A001678, A120803, A320154, A320160, A316624, A320169.

T[n_, k_] := T[n, k] = If[n==1, 1, If[k==0, 0, Sum[Sum[If[dJean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

Original entry on oeis.org

1, 1, 2, 6, 20, 90, 468, 2910, 20644, 165874, 1484344, 14653890, 158136988, 1852077284, 23394406084, 317018563806, 4587391330992, 70598570456104, 1151382852200680, 19835976878704628, 359963038816096924, 6863033015330999110, 137156667020252478684, 2867083618970831936826

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

For n > 1, also the number of balanced reduced multisystems whose atoms are an integer partition of n with at least one part > 1. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. - Gus Wiseman, Dec 31 2019

			The a(5) = 20 balanced reduced multisystems (with n written in place of 1^n):
  5  (14)  (23)  (113)      (122)      (1112)
                 ((1)(13))  ((1)(22))  ((1)(112))
                 ((3)(11))  ((2)(12))  ((2)(111))
                                       ((11)(12))
                                       ((1)(1)(12))
                                       ((1)(2)(11))
                                       (((1))((1)(12)))
                                       (((1))((2)(11)))
                                       (((2))((1)(11)))
                                       (((12))((1)(1)))
                                       (((11))((1)(2)))

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

The maximum-depth case is A000111.

Cf. A000311, A001055, A002846, A005121, A213427, A281118, A281119, A317145, A318812, A318846, A320154, A330474, A330679.

normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfact[n_]:=totfact[n]=1+Sum[totfact[Times@@Prime/@normize[f]],{f,Select[facs[n],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n, i, i==1), u=vector(n)); for(r=1, #v, u += v*sum(j=r, #v, (-1)^(j-r)*binomial(j-1, r-1)); v=EulerT(v)); u} \\ Andrew Howroyd, Dec 30 2019

a(n > 1) = A330679(n)/2. - Gus Wiseman, Dec 31 2019

Terms a(14) and beyond from Andrew Howroyd, Dec 30 2019

Terminology corrected by Gus Wiseman, Dec 31 2019

A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 3, 6, 9, 19, 31, 63, 110, 215, 391, 773, 1451, 2879, 5594, 11173, 22041, 44136, 87631, 175155, 348186, 694013, 1378911, 2743955, 5452833, 10853541, 21610732, 43122952, 86192274, 172753293, 347114772, 699602332, 1414033078, 2866580670, 5826842877, 11874508385

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

Also the number of balanced unlabeled phylogenetic rooted trees with n leaves.

			The a(1) = 1 through a(6) = 19 rooted trees:
  1  2     3      4           5            6
     (11)  (12)   (13)        (14)         (15)
           (111)  (22)        (23)         (24)
                  (112)       (113)        (33)
                  (1111)      (122)        (114)
                  ((11)(11))  (1112)       (123)
                              (11111)      (222)
                              ((11)(12))   (1113)
                              ((11)(111))  (1122)
                                           (11112)
                                           (111111)
                                           ((11)(13))
                                           ((11)(22))
                                           ((12)(12))
                                           ((11)(112))
                                           ((12)(111))
                                           ((11)(1111))
                                           ((111)(111))
                                           ((11)(11)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A316624, A320154, A320155, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[mps[Sort[labs]],Length[#1]>1&]}]];
Table[Sum[Length[Select[phy2[ptn],SameQ@@Length/@Position[#,_Integer]&]],{ptn,IntegerPartitions[n]}],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=vector(n, n, 1), v=vector(n)); while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(14) and beyond from Andrew Howroyd, Oct 25 2018

A119262 Number of B-trees of order infinity with n leaves, where a(n) = Sum_{k=1..floor(n/2)} a(k)C(n-k-1,n-2k) for n >= 2, with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 8, 14, 25, 46, 85, 158, 294, 548, 1022, 1908, 3567, 6683, 12556, 23669, 44781, 85046, 162122, 310157, 595322, 1146057, 2212004, 4278908, 8292738, 16097018, 31286456, 60873574, 118543329, 231009934, 450434739, 878687665

Offset: 0

Paul D. Hanna, May 11 2006

nonn

A B-tree of order m is an ordered tree such that every node has at most m children, the root has at least 2 children, every node except the root has 0 or at least m/2 children, all end-nodes are at the same level. This sequence is the limit of the B-trees as m --> infinity.
Starting with offset 2, the eigensequence of triangle A011973. - Gary W. Adamson, Jul 08 2012
Number of balanced series-reduced rooted plane trees with n leaves. A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root. - Gus Wiseman, Oct 07 2018

			A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 14*x^8 + ...
Series form:
A(x) = x + x^2/(1-x) + x^4/((1-x)*((1-x)-x^2)) + x^8/((1-x)*((1-x)-x^2)*((1-x)*((1-x)-x^2)-x^4)) + ... + x^(2^n)/D(n,x) + x^(2^(n+1))/[D(n,x)*(D(n,x)-x^(2^n))] + ...
Terms also satisfy the series:
x = x*(1-x) + x^2*(1-x^2)/(1+x) + x^3*(1-x^3)/(1+x)^2 + 2*x^4*(1-x^4)/(1+x)^3 + 3*x^5*(1-x^5)/(1+x)^4 + 5*x^6*(1-x^6)/(1+x)^5 + 8*x^7*(1-x^7)/(1+x)^6 + 14*x^8*(1-x^8)/(1+x)^7 + 25*x^9*(1-x^9)/(1+x)^8 + ... + a(n)*x^n*(1-x^n)/(1+x)^(n-1) + ...
From _Gus Wiseman_, Oct 07 2018: (Start)
The a(1) = 1 through a(7) = 8 balanced series-reduced rooted plane trees:
  o  (oo)  (ooo)  (oooo)      (ooooo)      (oooooo)        (ooooooo)
                  ((oo)(oo))  ((oo)(ooo))  ((oo)(oooo))    ((oo)(ooooo))
                              ((ooo)(oo))  ((ooo)(ooo))    ((ooo)(oooo))
                                           ((oooo)(oo))    ((oooo)(ooo))
                                           ((oo)(oo)(oo))  ((ooooo)(oo))
                                                           ((oo)(oo)(ooo))
                                                           ((oo)(ooo)(oo))
                                                           ((ooo)(oo)(oo))
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, B-Tree.

Cf. A092684 (similar recurrence); B-trees: A014535 (order 3), A037026 (order 4), A058521 (order 5).
Cf. A011973.
Cf. A000081, A000311, A001678, A048816, A079500, A120803, A316624, A320154, A320160, A320169.

nn=38;f[x_]:=Sum[a[n]x^n,{n,0,nn}];a[0]=0;sol=SolveAlways[0==Series[f[x]-x-f[x^2/(1-x)],{x,0,nn}],x];Table[a[n],{n,0,nn}]/.sol  (* Geoffrey Critzer, Mar 28 2013 *)

a(n)=if(n==0,0,if(n==1,1,sum(k=1,n\2,a(k)*binomial(n-k-1,n-2*k))))
for(n=1, 20, print1(a(n), ", "))

/* From: A(x) = x + A(x^2/(1-x)) */
{a(n)=local(A=x);for(i=1,n,A=x+subst(A,x,x^2/(1-x+x*O(x^n))));polcoeff(A,n)}
for(n=1, 20, print1(a(n), ", "))

/* From: x = Sum_{n>=1} a(n)*x^n*(1-x^n)/(1+x)^(n-1) */
a(n)=if(n==1, 1, -polcoeff(sum(k=1, n-1, a(k)*x^k*(1-x^k)/(1+x+x*O(x^n))^(k-1)), n))
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 31 2013

G.f. A(x) satisfies: A(x) = x + A(x^2/(1-x)).
G.f.: Sum_{n>=0} x^(2^n)/D(n,x) where D(0,x)=1, D(n+1,x) = D(n,x)*[D(n,x) - x^(2^n)].
G.f.: x = Sum_{n>=1} a(n) * x^n * (1-x^n) / (1+x)^(n-1). - Paul D. Hanna, Jul 31 2013
Conjecture: Let M_n be an n X n matrix whose elements are m_ij = 0 for i < j - 1, m_ij = -1 for i = j - 1, and m_ij = binomial(i - j, n - i) otherwise. Then a(n + 1) = Det(M_n). - Benedict W. J. Irwin, Apr 19 2017

A120803 Number of series-reduced balanced trees with n leaves.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 16, 20, 37, 47, 80, 111, 183, 256, 413, 591, 940, 1373, 2159, 3214, 5067, 7649, 12054, 18488, 29203, 45237, 71566, 111658, 176710, 276870, 437820, 687354, 1085577, 1705080, 2688285, 4221333, 6644088, 10425748

Offset: 1

Franklin T. Adams-Watters, Aug 18 2006

nonn

In other words, rooted trees with all leaves at the same level and no node having exactly one child; the order of children is not significant.

			From _Gus Wiseman_, Oct 07 2018: (Start)
The a(10) = 16 series-reduced balanced rooted trees:
  (oooooooooo)
  ((ooooo)(ooooo))
  ((oooo)(oooooo))
  ((ooo)(ooooooo))
  ((oo)(oooooooo))
  ((ooo)(ooo)(oooo))
  ((oo)(oooo)(oooo))
  ((oo)(ooo)(ooooo))
  ((oo)(oo)(oooooo))
  ((oo)(oo)(ooo)(ooo))
  ((oo)(oo)(oo)(oooo))
  ((oo)(oo)(oo)(oo)(oo))
  (((oo)(ooo))((oo)(ooo)))
  (((oo)(oo))((ooo)(ooo)))
  (((oo)(oo))((oo)(oooo)))
  (((oo)(oo))((oo)(oo)(oo)))
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A000669, A001003, A001678, A007059, A048816, A079500, A119262, A244925, A316624, A320154, A320160, A320169, A320179.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=vector(n), v=vector(n)); u[1]=1; while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 26 2018

Let s_0(n) = 1 if n = 1, 0 otherwise; s_{k+1}(n) = EULER(s_k)(n) - s_k(n), where EULER is the Euler transform. Then a_n = sum_k s_k(n). (s_k(n) is the number of such trees of height k.) Note that s_k(n) = 0 for n < 2^k.

A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

Original entry on oeis.org

1, 1, 2, 7, 48, 424

Offset: 0

Gus Wiseman, Dec 26 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(3) = 7 multisystems:
  {1,1,1}
  {1,1,2}
  {1,2,3}
  {{1},{1,1}}
  {{1},{1,2}}
  {{1},{2,3}}
  {{2},{1,1}}
Non-isomorphic representatives of the a(4) = 48 multisystems:
  {1,1,1,1}  {{1},{1,1,1}}    {{{1}},{{1},{1,1}}}
  {1,1,1,2}  {{1,1},{1,1}}    {{{1,1}},{{1},{1}}}
  {1,1,2,2}  {{1},{1,1,2}}    {{{1}},{{1},{1,2}}}
  {1,1,2,3}  {{1,1},{1,2}}    {{{1,1}},{{1},{2}}}
  {1,2,3,4}  {{1},{1,2,2}}    {{{1}},{{1},{2,2}}}
             {{1,1},{2,2}}    {{{1,1}},{{2},{2}}}
             {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,2}}    {{{1}},{{2},{1,1}}}
             {{1,2},{1,3}}    {{{1,2}},{{1},{1}}}
             {{1},{2,3,4}}    {{{1}},{{2},{1,2}}}
             {{1,2},{3,4}}    {{{1,2}},{{1},{2}}}
             {{2},{1,1,1}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{1},{1},{1,1}}  {{{1}},{{2},{3,4}}}
             {{1},{1},{1,2}}  {{{1,2}},{{3},{4}}}
             {{1},{1},{2,2}}  {{{2}},{{1},{1,1}}}
             {{1},{1},{2,3}}  {{{2}},{{1},{1,3}}}
             {{1},{2},{1,1}}  {{{2}},{{3},{1,1}}}
             {{1},{2},{1,2}}  {{{2,3}},{{1},{1}}}
             {{1},{2},{1,3}}
             {{1},{2},{3,4}}
             {{2},{3},{1,1}}

Labeled versions are A330475 (strongly normal) and A330655 (normal).
The case where the atoms are all different is A318813.
The case where the atoms are all equal is (also) A318813.
The labeled case of set partitions is A005121.
The labeled case of integer partitions is A330679.
The case of maximal depth is A330663.
The version where leaves are sets (as opposed to multisets) is A330668.
Cf. A000311, A000669, A001678, A002846, A004114, A007716, A048816, A213427, A306186, A320154, A320160, A330470, A330666.

A320169 Number of balanced enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 3, 6, 9, 20, 31, 70, 114, 243, 415, 961, 1603, 3564, 6559, 14913, 26630, 60037, 110160, 248859, 458445, 1001190, 1882350, 4220358, 7765303, 16822107, 32307240, 70081784, 133716083, 291788153, 561823990, 1230204229, 2396185727, 5176454708, 10220127290

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

An enriched p-tree of weight n is either the number n itself or a finite sequence of enriched p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(6) = 20 balanced enriched p-trees:
  1  2     3      4           5            6
     (11)  (21)   (22)        (32)         (33)
           (111)  (31)        (41)         (42)
                  (211)       (221)        (51)
                  (1111)      (311)        (222)
                  ((11)(11))  (2111)       (321)
                              (11111)      (411)
                              ((21)(11))   (2211)
                              ((111)(11))  (3111)
                                           (21111)
                                           (111111)
                                           ((21)(21))
                                           ((22)(11))
                                           ((31)(11))
                                           ((111)(21))
                                           ((21)(111))
                                           ((211)(11))
                                           ((111)(111))
                                           ((1111)(11))
                                           ((11)(11)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

Cf. A316624, A320154, A320155, A320160, A320179.

eptrs[n_]:=Prepend[Join@@Table[Tuples[eptrs/@p],{p,Rest[IntegerPartitions[n]]}],n];
Table[Length[Select[eptrs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,12}]

seq(n)={my(p=x/(1-x) + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 26 2018

A320179 Regular 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.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 6, 1, 0, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 0, 0, 1, 11, 4, 0, 0, 0, 0, 0, 0, 0, 1, 13, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 23, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 07 2018

			Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  1  0  0
  0  1  3  0  0  0
  0  1  3  0  0  0  0
  0  1  6  1  0  0  0  0
  0  1  7  1  0  0  0  0  0
  0  1 11  4  0  0  0  0  0  0
  0  1 13  6  0  0  0  0  0  0  0
  0  1 20 16  0  0  0  0  0  0  0  0
  0  1 23 23  0  0  0  0  0  0  0  0  0
  0  1 33 46  0  0  0  0  0  0  0  0  0  0
The T(10,3) = 4 rooted trees:
   (((oo)(oo))((oo)(oooo)))
   (((oo)(oo))((ooo)(ooo)))
   (((oo)(ooo))((oo)(ooo)))
  (((oo)(oo))((oo)(oo)(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A120803. Third column is A083751. An irregular version is A320221.
Cf. A000669, A001678, A048816, A079500, A119262, A244925, A316655, 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&]}]];
Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,14},{k,0,n-1}]

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); vector(n, n, Vecrev(v[n], n))}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020

A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

Original entry on oeis.org

1, 1, 1, 4, 11, 41, 162, 1030, 7205, 55522, 442443, 3810852, 35272030, 351697516, 3735838550, 42719792640, 529195988635, 7128835815387, 103651381499810, 1610812109555323, 26489497655582729, 457497408108551450, 8248899117402701046, 154624472715479106919

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

			The a(1) = 1 through a(5) = 11 rooted trees:
  1  (12)  (123)    (1234)      (12345)
                  ((12)(34))  ((12)(345))
                  ((13)(24))  ((13)(245))
                  ((14)(23))  ((14)(235))
                              ((15)(234))
                              ((23)(145))
                              ((24)(135))
                              ((25)(134))
                              ((34)(125))
                              ((35)(124))
                              ((45)(123))

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

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A320154, A320160, A316624, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[sps[Sort[labs]],Length[#1]>1&]}]];
Table[Length[Select[phy2[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,7}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

E.g.f. A(x) satisfies A(x) = x + A(exp(x)-x-1). - Ira M. Gessel, Nov 17 2021

Terms a(10) and beyond from Andrew Howroyd, Oct 26 2018

cp's OEIS Frontend

A079500 Number of compositions of the integer n in which the first part is >= the other parts.

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A048816 Number of rooted trees with n nodes with every leaf at the same height.

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A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

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A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

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A119262 Number of B-trees of order infinity with n leaves, where a(n) = Sum_{k=1..floor(n/2)} a(k)*C(n-k-1,n-2*k) for n >= 2, with a(0)=0, a(1)=1.

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A120803 Number of series-reduced balanced trees with n leaves.

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A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

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A119262 Number of B-trees of order infinity with n leaves, where a(n) = Sum_{k=1..floor(n/2)} a(k)C(n-k-1,n-2k) for n >= 2, with a(0)=0, a(1)=1.