A048816 -id:A048816 - cp's OEIS frontend

A000311 Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n.

0, 1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912, 282137824, 6939897856, 188666182784, 5617349020544, 181790703209728, 6353726042486272, 238513970965257728, 9571020586419012608, 408837905660444010496, 18522305410364986906624

Offset: 0

N. J. A. Sloane

a(n) is the number of labeled series-reduced rooted trees with n leaves (root has degree 0 or >= 2); a(n-1) = number of labeled series-reduced trees with n leaves. Also number of series-parallel networks with n labeled edges, divided by 2.
A total partition of n is essentially what is meant by the first part of the previous line: take the numbers 12...n, and partition them into at least two blocks. Partition each block with at least 2 elements into at least two blocks. Repeat until only blocks of size 1 remain. (See the reference to Stanley, Vol. 2.) - N. J. A. Sloane, Aug 03 2016
Polynomials with coefficients in triangle A008517, evaluated at 2. - Ralf Stephan, Dec 13 2004
Row sums of unsigned A134685. - Tom Copeland, Oct 11 2008
Row sums of A134991, which contains an e.g.f. for this sequence and its compositional inverse. - Tom Copeland, Jan 24 2018
From Gus Wiseman, Dec 28 2019: (Start)
Also the number of singleton-reduced phylogenetic trees with n labels. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) nonempty sets. It is singleton-reduced if no non-leaf node covers only singleton branches. For example, the a(4) = 26 trees are:
  {1,2,3,4}  {{1},{2},{3,4}}  {{1},{2,3,4}}
             {{1},{2,3},{4}}  {{1,2},{3,4}}
             {{1,2},{3},{4}}  {{1,2,3},{4}}
             {{1},{2,4},{3}}  {{1,2,4},{3}}
             {{1,3},{2},{4}}  {{1,3},{2,4}}
             {{1,4},{2},{3}}  {{1,3,4},{2}}
                              {{1,4},{2,3}}
                              {{{1},{2,3}},{4}}
                              {{{1,2},{3}},{4}}
                              {{1},{{2},{3,4}}}
                              {{1},{{2,3},{4}}}
                              {{{1},{2,4}},{3}}
                              {{{1,2},{4}},{3}}
                              {{1},{{2,4},{3}}}
                              {{{1,3},{2}},{4}}
                              {{{1},{3,4}},{2}}
                              {{{1,3},{4}},{2}}
                              {{{1,4},{2}},{3}}
                              {{{1,4},{3}},{2}}
(End)

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 26*x^4/4! + 236*x^5/5! + 2752*x^6/6! + ...
where exp(A(x)) = 1 - x + 2*A(x), and thus
Series_Reversion(A(x)) = x - x^2/2! - x^3/3! - x^4/4! - x^5/5! - x^6/6! + ...
O.g.f.: G(x) = x + x^2 + 4*x^3 + 26*x^4 + 236*x^5 + 2752*x^6 + 39208*x^7 + ...
where
G(x) = x/2 + x/(2*(2-x)) + x/(2*(2-x)*(2-2*x)) + x/(2*(2-x)*(2-2*x)*(2-3*x)) + x/(2*(2-x)*(2-2*x)*(2-3*x)*(2-4*x)) + x/(2*(2-x)*(2-2*x)*(2-3*x)*(2-4*x)*(2-5*x)) + ...
From _Gus Wiseman_, Dec 28 2019: (Start)
A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes. The a(4) = 26 series-reduced rooted trees with 4 labeled leaves are the following. Each bracket (...) corresponds to a non-leaf node.
  (1234)  ((12)34)  ((123)4)
          (1(23)4)  (1(234))
          (12(34))  ((124)3)
          (1(24)3)  ((134)2)
          ((13)24)  (((12)3)4)
          ((14)23)  ((1(23))4)
                    ((12)(34))
                    (1((23)4))
                    (1(2(34)))
                    (((12)4)3)
                    ((1(24))3)
                    (1((24)3))
                    (((13)2)4)
                    ((13)(24))
                    (((13)4)2)
                    ((1(34))2)
                    (((14)2)3)
                    ((14)(23))
                    (((14)3)2)
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 224.
J. Felsenstein, Inferring phyogenies, Sinauer Associates, 2004; see p. 25ff.
L. R. Foulds and R. W. Robinson, Enumeration of phylogenetic trees without points of degree two. Ars Combin. 17 (1984), A, 169-183. Math. Rev. 85f:05045
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 197.
E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see "total partitions", Example 5.2.5, Equation (5.27), and also Fig. 5-3 on page 14. See also the Notes on page 66.

N. J. A. Sloane, Table of n, a(n) for n = 0..375 [Shortened file because terms grow rapidly: see Sloane link below for additional terms]
Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin, Mickaël Maazoun, and Adeline Pierrot, Random cographs: Brownian graphon limit and asymptotic degree distribution, arXiv:1907.08517 [math.CO], 2019.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
A. Blass, N. Dobrinen, and D. Raghavan, The next best thing to a P-point, arXiv preprint arXiv:1308.3790 [math.LO], 2013.
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. MR0891613 (89a:05009). See p. 155 and 159.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. Carlitz and J. Riordan, The number of labeled two-terminal series-parallel networks, Duke Math. J. 23 (1956), 435-445 (the sequence called {A_n}).
Tom Copeland, Comments on A000311
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016.
J. Felsenstein, The number of evolutionary trees, Systematic Zoology, 27 (1978), 27-33. (Annotated scanned copy)
J. Felsenstein, The number of evolutionary trees, Systematic Biology, 27 (1978), pp. 27-33, 1978.
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
Philippe Flajolet, A Problem in Statistical Classification Theory
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 129
Daniel L. Geisler, Combinatorics of Iterated Functions
M. D. Hendy, C. H. C. Little, David Penny, Comparing trees with pendant vertices labelled, SIAM J. Appl. Math. 44 (5) (1984) Table 1
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 69
D. Jackson, A. Kempf, and A. Morales, A robust generalization of the Legendre transform for QFT, arXiv:1612.0046 [hep-th], 2017.
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Vladimir V. Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
Shi-Mei Ma, Jun-Ying Liu, Jean Yeh, and Yeong-Nan Yeh, Eulerian-type polynomials over Stirling permutations and box sorting algorithm, arXiv:2506.16438 [math.CO], 2025. See p. 5.
Dragan Mašulović, Big Ramsey spectra of countable chains, arXiv:1912.03022 [math.CO], 2019.
Arnau Mir, Francesc Rossello, and Lucia Rotger, Sound Colless-like balance indices for multifurcating trees, arXiv:1805.01329 [q-bio.PE], 2018.
J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
F. Murtagh, Counting dendrograms: a survey, Discrete Appl. Math., 7 (1984), 191-199.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
P. Regner, Phylogenetic Trees: Selected Combinatorial Problems, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, pp. 50-59.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1970
J. Riordan, The blossoming of Schröder's fourth problem, Acta Math., 137 (1976), 1-16.
E. Schröder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy]
N. J. A. Sloane, Table of n, a(n) for n = 0..500
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95. [_Tom Copeland_, Dec 20 2018]
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 2, 12.
Index entries for sequences related to trees
Index entries for sequences related to rooted trees
Index entries for "core" sequences
Index entries for sequences mentioned in Moon (1987)
Index entries for sequences related to parenthesizing

Row sums of A064060 and A134991.
The unlabeled version is A000669.
Unlabeled phylogenetic trees are A141268.
The node-counting version is A060356, with unlabeled version A001678.
Phylogenetic trees with n labels are A005804.
Chains of set partitions are A005121, with maximal version A002846.
Inequivalent leaf-colorings of series-reduced rooted trees are A318231.
Cf. A001003, A007827, A005805, A006351, A000084.
For n >= 2, A000311(n) = A006351(n)/2 = A005640(n)/2^(n+1).
Cf. A000110, A000669 = unlabeled hierarchies, A119649.
Cf. A000670, A135494, A134685.
Cf. A048816, A196545, A213427, A316651, A316652, A330626, A330627.
Cf. A000169, A042977, A133314, A134685, A269939.

M:=499; a:=array(0..500); a[0]:=0; a[1]:=1; a[2]:=1; for n from 0 to 2 do lprint(n,a[n]); od: for n from 2 to M do a[n+1]:=(n+2)*a[n]+2*add(binomial(n,k)*a[k]*a[n-k+1],k=2..n-1); lprint(n+1,a[n+1]); od:
Order := 50; t1 := solve(series((exp(A)-2*A-1),A)=-x,A); A000311 := n-> n!*coeff(t1,x,n);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(combinat[multinomial](n, n-i*j, i$j)/j!*
      a(i)^j*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n, n-1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 28 2016
# faster program:
b:= proc(n, i) option remember;
    `if`(i=0 and n=0, 1, `if`(i<=0 or i>n, 0,
    i*b(n-1, i) + (n+i-1)*b(n-1, i-1))) end:
a:= n -> `if`(n<2, n, add(b(n-1, i), i=0..n-1)):
seq(a(n), n=0..40);  # Peter Luschny, Feb 15 2021

nn = 19; CoefficientList[ InverseSeries[ Series[1+2a-E^a, {a, 0, nn}], x], x]*Range[0, nn]! (* Jean-François Alcover, Jul 21 2011 *)
a[ n_] := If[ n < 1, 0, n! SeriesCoefficient[ InverseSeries[ Series[ 1 + 2 x - Exp[x], {x, 0, n}]], n]]; (* Michael Somos, Jun 04 2012 *)
a[n_] := (If[n < 2,n,(column = ConstantArray[0, n - 1]; column[[1]] = 1; For[j = 3, j <= n, j++, column = column * Flatten[{Range[j - 2], ConstantArray[0, (n - j) + 1]}] + Drop[Prepend[column, 0], -1] * Flatten[{Range[j - 1, 2*j - 3], ConstantArray[0, n - j]}];]; Sum[column[[i]], {i, n - 1}]  )]); Table[a[n], {n, 0, 20}] (* Peter Regner, Oct 05 2012, after a formula by Felsenstein (1978) *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&,j]]]/j!*a[i]^j *b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<2, n, b[n, n-1]]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 07 2016, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[sps[m],1Gus Wiseman, Dec 28 2019 *)
(* Lengthy but easy to follow *)
  lead[, n /; n < 3] := 0
  lead[h_, n_] := Module[{p, i},
        p = Position[h, {_}];
        Sum[MapAt[{#, n} &, h, p[[i]]], {i, Length[p]}]
        ]
  follow[h_, n_] := Module[{r, i},
        r = Replace[Position[h, {_}], {a__} -> {a, -1}, 1];
        Sum[Insert[h, n, r[[i]]], {i, Length[r]}]
        ]
  marry[, n /; n < 3] := 0
  marry[h_, n_] := Module[{p, i},
        p = Position[h, _Integer];
        Sum[MapAt[{#, n} &, h, p[[i]]], {i, Length[p]}]
        ]
  extend[a_ + b_, n_] := extend[a, n] + extend[b, n]
  extend[a_, n_] := lead[a, n] + follow[a, n] + marry[a, n]
  hierarchies[1] := hierarchies[1] = extend[hier[{}], 1]
  hierarchies[n_] := hierarchies[n] = extend[hierarchies[n - 1], n] (* Daniel Geisler, Aug 22 2022 *)

a(n):=if n=1 then 1 else sum((n+k-1)!*sum(1/(k-j)!*sum((2^i*(-1)^(i)*stirling2(n+j-i-1,j-i))/((n+j-i-1)!*i!),i,0,j),j,1,k),k,1,n-1); /* Vladimir Kruchinin, Jan 28 2012 */

{a(n) = local(A); if( n<0, 0, for( i=1, n, A = Pol(exp(A + x * O(x^i)) - A + x - 1)); n! * polcoeff(A, n))}; /* Michael Somos, Jan 15 2004 */

{a(n) = my(A); if( n<0, 0, A = O(x); for( i=1, n, A = intformal( 1 / (1 + x - 2*A))); n! * polcoeff(A, n))}; /* Michael Somos, Oct 25 2014 */

{a(n) = n! * polcoeff(serreverse(1+2*x - exp(x +x^2*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 27 2014

\p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*x/prod(k=0, n, 2 - k*x + O(x^31))))}
for(n=0, 25, print1(if(n<1,0,round(A[n])),", ")) \\ Paul D. Hanna, Oct 27 2014

from functools import lru_cache
from math import comb
@lru_cache(maxsize=None)
def A000311(n): return n if n <= 1 else -(n-1)*A000311(n-1)+comb(n,m:=n+1>>1)*(0 if n&1 else A000311(m)**2) + (sum(comb(n,i)*A000311(i)*A000311(n-i) for i in range(1,m))<<1) # Chai Wah Wu, Nov 10 2022

E.g.f. A(x) satisfies exp A(x) = 2*A(x) - x + 1.
a(0)=0, a(1)=a(2)=1; for n >= 2, a(n+1) = (n+2)*a(n) + 2*Sum_{k=2..n-1} binomial(n, k)*a(k)*a(n-k+1).
a(1)=1; for n>1, a(n) = -(n-1) * a(n-1) + Sum_{k=1..n-1} binomial(n, k) * a(k) * a(n-k). - Michael Somos, Jun 04 2012
From the umbral operator L in A135494 acting on x^n comes, umbrally, (a(.) + x)^n = (n * x^(n-1) / 2) - (x^n / 2) + Sum_{j>=1} j^(j-1) * (2^(-j) / j!) * exp(-j/2) * (x + j/2)^n giving a(n) = 2^(-n) * Sum_{j>=1} j^(n-1) * ((j/2) * exp(-1/2))^j / j! for n > 1. - Tom Copeland, Feb 11 2008
Let h(x) = 1/(2-exp(x)), an e.g.f. for A000670, then the n-th term of A000311 is given by ((h(x)*d/dx)^n)x evaluated at x=0, i.e., A(x) = exp(x*a(.)) = exp(x*h(u)*d/du) u evaluated at u=0. Also, dA(x)/dx = h(A(x)). - Tom Copeland, Sep 05 2011 (The autonomous differential eqn. here is also on p. 59 of Jones. - Tom Copeland, Dec 16 2019)
A134991 gives (b.+c.)^n = 0^n, for (b_n)=A000311(n+1) and (c_0)=1, (c_1)=-1, and (c_n)=-2* A000311(n) = -A006351(n) otherwise. E.g., umbrally, (b.+c.)^2 = b_2*c_0 + 2 b_1*c_1 + b_0*c_2 =0. - Tom Copeland, Oct 19 2011
a(n) = Sum_{k=1..n-1} (n+k-1)!*Sum_{j=1..k} (1/(k-j)!)*Sum_{i=0..j} 2^i*(-1)^i*Stirling2(n+j-i-1, j-i)/((n+j-i-1)!*i!), n>1, a(0)=0, a(1)=1. - Vladimir Kruchinin, Jan 28 2012
Using L. Comtet's identity and D. Wasserman's explicit formula for the associated Stirling numbers of second kind (A008299) one gets: a(n) = Sum_{m=1..n-1} Sum_{i=0..m} (-1)^i * binomial(n+m-1,i) * Sum_{j=0..m-i} (-1)^j * ((m-i-j)^(n+m-1-i))/(j! * (m-i-j)!). - Peter Regner, Oct 08 2012
G.f.: x/Q(0), where Q(k) = 1 - k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: x*Q(0), where Q(k) = 1 - x*(k+1)/(x*(k+1) - (1-k*x)*(1-x-k*x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2013
a(n) ~ n^(n-1) / (sqrt(2) * exp(n) * (2*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Jan 05 2014
E.g.f. A(x) satisfies d/dx A(x) = 1 / (1 + x - 2 * A(x)). - Michael Somos, Oct 25 2014
O.g.f.: Sum_{n>=0} x / Product_{k=0..n} (2 - k*x). - Paul D. Hanna, Oct 27 2014
E.g.f.: (x - 1 - 2 LambertW(-exp((x-1)/2) / 2)) / 2. - Vladimir Reshetnikov, Oct 16 2015 (This e.g.f. is given in A135494, the entry alluded to in my 2008 formula, and in A134991 along with its compositional inverse. - Tom Copeland, Jan 24 2018)
a(0) = 0, a(1) = 1; a(n) = n! * [x^n] exp(Sum_{k=1..n-1} a(k)*x^k/k!). - Ilya Gutkovskiy, Oct 17 2017
a(n+1) = Sum_{k=0..n} A269939(n, k) for n >= 1. - Peter Luschny, Feb 15 2021

Name edited by Gus Wiseman, Dec 28 2019

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

Original entry on oeis.org

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

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

Original entry on oeis.org

0, 1, 0, 3, 4, 65, 306, 4207, 38424, 573057, 7753510, 134046671, 2353898196, 47602871329, 1013794852266, 23751106404495, 590663769125296, 15806094859299329, 448284980183376078, 13515502344669830287

Offset: 0

Vladeta Jovovic, Apr 01 2001

Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - Gus Wiseman, Jan 20 2020

			From _Gus Wiseman_, Dec 31 2019: (Start)
Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
  1[2,3[4,5[6,7]]]
  1[2,3[4,5,6,7]]
  1[2[3,4],5[6,7]]
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A052871, A060313.
Cf. A008297.
Column k=0 of A231602.
The unlabeled version is A001678(n + 1).
The case where the root is fixed is A108919.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees with labeled leaves are A000311.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.

List([0..20],n->Sum([1..n],k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1,k-1)*k^(k-1))); # Muniru A Asiru, Feb 19 2018

seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # G. C. Greubel, Mar 16 2020

CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];
Array[a,10] (* Gus Wiseman, Dec 31 2019 *)

{ for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009

my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ G. C. Greubel, Feb 19 2018

a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)*A058863(k). - Vladeta Jovovic, Sep 17 2003
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Nov 27 2012
a(n) = n * A108919(n). - Gus Wiseman, Dec 31 2019

A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 3, 2, 1, 1, 0, 1, 4, 5, 3, 2, 1, 1, 0, 1, 7, 7, 6, 3, 2, 1, 1, 0, 1, 8, 12, 8, 6, 3, 2, 1, 1, 0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1, 0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1, 0, 1, 21, 42, 39, 26, 17, 9, 6, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Jul 08 2014

			The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height are:
  :    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  :
  :         :             :       :     :
  : ---1--- : -----2----- : --3-- : -4- :
Thus row 5 = [0, 1, 2, 1, 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  1,  1;
  0, 1,  2,  1,  1;
  0, 1,  2,  2,  1,  1;
  0, 1,  4,  3,  2,  1, 1;
  0, 1,  4,  5,  3,  2, 1, 1;
  0, 1,  7,  7,  6,  3, 2, 1, 1;
  0, 1,  8, 12,  8,  6, 3, 2, 1, 1;
  0, 1, 12, 18, 15,  9, 6, 3, 2, 1, 1;
  0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1;
  ...

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

Columns k=0-10 give: A000007(n-1), A000012 (for n>0), A002865(n-1) (for n>2), A048808, A048809, A048810, A048811, A048812, A048813, A048814, A048815.
T(2n+1,n) gives A074045.
Row sums give A048816.

with(numtheory):
T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0,
      add(add(`if`(d

T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[dJean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

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

cp's OEIS Frontend

A000311 Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n.

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

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A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

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A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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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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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.