A292555 -id:A292555 - cp's OEIS frontend

A001190 Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).

0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, 127912, 293547, 676157, 1563372, 3626149, 8436379, 19680277, 46026618, 107890609, 253450711, 596572387, 1406818759, 3323236238, 7862958391, 18632325319, 44214569100, 105061603969

Offset: 0

N. J. A. Sloane

Also number of n-node binary rooted trees (every node has outdegree <= 2) where root has degree 0 (only for n=1) or 1.
a(n+1) is the number of rooted trees with n nodes where the outdegree of every node is <= 2, see example. These trees are obtained by removing the root of the trees in the comment above. - Joerg Arndt, Jun 29 2014
Number of interpretations of x^n (or number of ways to insert parentheses) when multiplication is commutative but not associative. E.g., a(4) = 2: x(x*x^2) and x^2*x^2. a(5) = 3: (x*x^2)x^2, x(x*x*x^2) and x(x^2*x^2). [If multiplication is non-commutative then the answer is A000108(n-1). - Jianing Song, Apr 29 2022]
Number of ways to place n stars in a single bound stable hierarchical multiple star system; i.e., taking only the configurations from A003214 where all stars are included in single outer parentheses. - Piet Hut, Nov 07 2003
Number of colorations of Kn (complete graph of order n) with n-1 colors such that no triangle is three-colored. Two edge-colorations C1 and C2 of G are isomorphic iff exists an automorphism f (isomorphism between G an G) such that: f sends same-colored edges of C1 on same-colored edges of C2 and f^(-1) sends same-colored edges of C2 on same-colored edges of C1. - Abraham Gutiérrez, Nov 12 2012
For n>1, a(n) is the number of (not necessarily distinct) unordered pairs of free unlabeled trees having a total of n nodes. See the first entry in formula section. - Geoffrey Critzer, Nov 09 2014
Named after the English mathematician Ivor Etherington (1908-1994) and the Scottish mathematician Joseph Wedderburn (1882-1948). - Amiram Eldar, May 29 2021

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + ...
From _Joerg Arndt_, Jun 29 2014: (Start)
The a(6+1) = 11 rooted trees with 6 nodes as described in the comment are:
:           level sequence       outdegrees (dots for zeros)
:     1:  [ 0 1 2 3 4 5 ]    [ 1 1 1 1 1 . ]
:  O--o--o--o--o--o
:
:     2:  [ 0 1 2 3 4 4 ]    [ 1 1 1 2 . . ]
:  O--o--o--o--o
:           .--o
:
:     3:  [ 0 1 2 3 4 3 ]    [ 1 1 2 1 . . ]
:  O--o--o--o--o
:        .--o
:
:     4:  [ 0 1 2 3 4 2 ]    [ 1 2 1 1 . . ]
:  O--o--o--o--o
:     .--o
:
:     5:  [ 0 1 2 3 4 1 ]    [ 2 1 1 1 . . ]
:  O--o--o--o--o
:  .--o
:
:     6:  [ 0 1 2 3 3 2 ]    [ 1 2 2 . . . ]
:  O--o--o--o
:        .--o
:     .--o
:
:     7:  [ 0 1 2 3 3 1 ]    [ 2 1 2 . . . ]
:  O--o--o--o
:        .--o
:  .--o
:
:     8:  [ 0 1 2 3 2 3 ]    [ 1 2 1 . 1 . ]
:  O--o--o--o
:     .--o--o
:
:     9:  [ 0 1 2 3 2 1 ]    [ 2 2 1 . . . ]
:  O--o--o--o
:     .--o
:  .--o
:
:    10:  [ 0 1 2 3 1 2 ]    [ 2 1 1 . 1 . ]
:  O--o--o--o
:  .--o--o
:
:    11:  [ 0 1 2 2 1 2 ]    [ 2 2 . . 1 . ]
:  O--o--o
:     .--o
:  .--o--o
:
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 307.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 55.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
A. Gutiérrez-Sánchez, Shen-colored tournaments, thesis, UNAM, 2012.
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).
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.52.
Richard P. Stanley, Catalan Numbers, Cambridge, 2015, p. 133.

Alois P. Heinz, Table of n, a(n) for n = 0..2545 (first 201 terms from T. D. Noe)
R. Arratia, S. Garibaldi, and A. W. Hales, The van den Berg--Kesten--Reimer inequality for infinite spaces, arXiv preprint arXiv:1508.05337 [math.PR], 2015.
Yu Hin (Gary) Au, Fatemeh Bagherzadeh, and Murray R. Bremner, Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube, arXiv:1903.00813 [math.CO], 2019.
F. Bagherzadeh, M. R. Bremner, and S. Madariaga, Jordan trialgebras and post-Jordan algebras, arXiv:1611.01214 [math.RA], 2016.
Nils Berglund and Yvain Bruned, BPHZ renormalisation and vanishing subcriticality limit of the fractional Phi_d^3 model, arXiv:1907.13028 [math.PR], 2019.
Nils Berglund and Christian Kuehn, Model Spaces of Regularity Structures for Space-Fractional SPDEs, Journal of Statistical Physics, Springer Verlag, 2017, 168 (2), pp. 331-368; HAL Id: hal-01432157.
Mayfawny Bergmann, Efficiency of Lossless Compression of a Binary Tree via its Minimal Directed Acyclic Graph Representation. Rose-Hulman Undergraduate Mathematics Journal: Vol. 15: Iss. 2, Article 1 (2014).
Sara Billey, Matjaz Konvalinka, and Frederick A Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.
Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On trees, tanglegrams, and tangled chains, hal-02173394 [math.CO], 2020.
Henry Bottomley, Illustration of initial terms.
M. Bremner, S. Madariaga, and L. A. Peresi, Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions, arXiv:1407.3810 [math.RA], 2014.
Nicolas Broutin and Philippe Flajolet, The distribution of height and diameter in random non-plane binary trees, Random Struct. Algorithms 41, No. 2, 215-252 (2012).
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155-183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
Lorenzo Cappello and Julia A. Palacios, Sequential importance sampling for multi-resolution Kingman-Tajima coalescent counting, arXiv:1902.05527 [stat.AP], 2019.
Sean Cleary, M. Fischer, R. C. Griffiths, and R. Sainudiin, Some distributions on finite rooted binary trees, UCDMS Research Report NO. UCDMS2015/2, School of Mathematics and Statistics, University of Canterbury, Christchurch, NZ, 2015.
S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Enumeration of constitutional isomers of polyenes, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261.
N. G. de Bruijn and D. A. Klarner, Multisets of aperiodic cycles, SIAM J. Algebraic Discrete Methods 3 (1982), no. 3, 359-368. MR0666861(84i:05008). See p. 367. - _N. J. A. Sloane_, Mar 25 2014
Jimmy Devillet and Bruno Teheux, Associative, idempotent, symmetric, and order-preserving operations on chains, arXiv:1805.11936 [math.RA], 2018.
Luc Devroye, Michael R. Doboli, Noah A. Rosenberg, and Stephan Wagner, Tree height and the asymptotic mean of the Colijn-Plazzotta rank of unlabeled binary rooted trees, arXiv:2409.18956 [math.CO], 2024. See p. 21.
Filippo Disanto and Thomas Wiehe, Some combinatorial problems on binary rooted trees occurring in population genetics, arXiv preprint arXiv:1112.1295 [math.CO], 2011-2012.
Michael R. Doboli, Hsien-Kuei Hwang, and Noah A. Rosenberg, Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves, In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 18:1-18:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2024).
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39 and 153.
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162. [Annotated scanned copy]
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi.
A. Erdelyi and I. M. H. Etherington, Some problems of non-associative combinations (II), Edinburgh Math. Notes, 32 (1940), pp. vii-xiv.
V. Fack, S. Lievens and J. Van der Jeugt, On the diameter of the rotation graph of binary coupling trees. Discrete Math. 245 (2002), no. 1-3, 1--18. MR1887046 (2003i:05047).
Steven R. Finch, Otter's Tree Enumeration Constants. [Broken link]
Steven R. Finch, Otter's Tree Enumeration Constants. [Wayback Machine]
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 72
J. N. Franklin and S. W. Golomb, A Function-Theoretic Approach to the Study of Nonlinear Recurring Sequences, Pacific J. Math., Vol. 56, p. 467, 1975.
Ira M. Gessel, Counting tanglegrams with species, arXiv:1509.03867 [math.CO], 2020.
Piet Hut, Home Page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 43
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 45
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph.D. Dissertation, University of South Carolina, 2012.
M. Konvalinka and S. Wagner, The shape of random tanglegrams, arXiv preprint arXiv:1512.01168 [math.CO], 2015.
A. Ledda, G. Achaz, T. Wiehe and L. Ferretti, Decomposing the site frequency spectrum: the impact of tree topology on neutrality tests, arXiv preprint arXiv:1510.06748 [q-bio.PE], 2015.
Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, Toric Richardson varieties of Catalan type and Wedderburn-Etherington numbers, arXiv:2105.12274 [math.AG], 2021.
F. Murtagh, Counting dendrograms: a survey, Discrete Applied Mathematics, 7 (1984), 191-199.
C. D. Olds, Problem 4277, Amer. Math. Monthly, 56 (1949), 697-699.
C. D. Olds (Proposer) and H. W. Becker (Discussion), Problem 4277, Amer. Math. Monthly 56 (1949), 697-699. [Annotated scanned copy]
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
David Serena and William J. Buchanan, Equivalence Classes Induced by Binary Tree Isomorphism -- Generating Functions, arXiv:2503.02663 [math.CO], 2025. See p. 5.
Chloe E. Shiff and Noah A. Rosenberg, Enumeration of rooted binary perfect phylogenies, arXiv:2410.14915 [q-bio.PE], 2024. See pp. 2, 5, 9.
F. Sievers, G. M. Hughes, and D. G. Higgins, Systematic Exploration of Guide-Tree Topology Effects for Small Protein Alignments, BMC Bioinformatics 2014, 15:338 (Mentions A001190).
J. H. M. Wedderburn, The functional equation g(x^2) = 2ax + [g(x)]^2, Ann. Math., 24 (1922-23), 121-140.
Eric Weisstein's World of Mathematics, Weakly Binary Tree.
Eric Weisstein's World of Mathematics, Strongly Binary Tree.
Wikipedia, Wedderburn-Etherington numbers.
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for sequences related to parenthesizing

Cf. A000108, A001699, A002658, A003214, A006894, A006961, A088325.
Cf. A086317, A086318, A240943.
Cf. A292553, A292554, A292555, A292556.
Column k=2 of A292085 and of A299038.
Column k=1 of A319539 and of A319541.

A001190 := proc(n) option remember; local s,k; if n<=1 then RETURN(n); elif n <=3 then RETURN(1); else s := 0; if n mod 2 = 0 then s := A001190(n/2)*(A001190(n/2)+1)/2; for k from 1 to n/2-1 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); else for k from 1 to (n-1)/2 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); fi; fi; end;
N := 40: G001190 := add(A001190(n)*x^n,n=0..N);
spec := [S,{S=Union(Z,Prod(Z,Set(S,card=2)))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
# alternative Maple program:
a:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(a(n/2)))+add(a(i)*a(n-i), i=1..n/2))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 28 2017

terms = 35; A[] = 0; Do[A[x] = x + (1/2)*(A[x]^2 + A[x^2]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jul 22 2011, updated Jan 10 2018 *)
a[n_?OddQ] := a[n] = Sum[a[k]*a[n-k], {k, 1, (n-1)/2}]; a[n_?EvenQ] := a[n] = Sum[a[k]*a[n-k], {k, 1, n/2-1}] + (1/2)*a[n/2]*(1+a[n/2]); a[0]=0; a[1]=1; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 13 2012, after recurrence formula *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Nest[ 1 - Sqrt[1 - 2 x - (# /. x -> x^2)] &, 0, BitLength @ n], {x, 0, n}]]; (* Michael Somos, Apr 25 2013 *)

{a(n) = local(A, m); if( n<0, 0, m=1; A = O(x); while( m<=n, m*=2; A = 1 - sqrt(1 - 2*x - subst(A, x, x^2))); polcoeff(A, n))}; /* Michael Somos, Sep 06 2003 */

{a(n) = local(A); if( n<4, n>0, A = vector(n, i, 1); for( i=4, n, A[i] = sum( j=1, (i-1)\2, A[j] * A[i-j]) + if( i%2, 0, A[i/2] * (A[i/2] + 1)/2)); A[n])}; /* Michael Somos, Mar 25 2006 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A001190(n):
    if n <= 1: return n
    m = n//2 + n % 2
    return sum(A001190(i+1)*A001190(n-1-i) for i in range(m-1)) + (1 - n % 2)*A001190(m)*(A001190(m)+1)//2 # Chai Wah Wu, Jan 14 2022

G.f. satisfies A(x) = x + (1/2)*(A(x)^2 + A(x^2)) [de Bruijn and Klarner].
G.f. also satisfies A(x) = 1 - sqrt(1 - 2*x - A(x^2)). - Michael Somos, Sep 06 2003
a(2n-1) = a(1)a(2n-2) + a(2)a(2n-3) + ... + a(n-1)a(n), a(2n) = a(1)a(2n-1) + a(2)a(2n-2) + ... + a(n-1)a(n+1) + a(n)(a(n)+1)/2.
Given g.f. A(x), then B(x) = -1 + A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = (u^2 + v)^2 + 2*(v^2 + w). - Michael Somos, Oct 22 2006
The radius of convergence of the g.f. is A240943 = 1/A086317 ~ 0.4026975... - Jean-François Alcover, Jul 28 2014, after Steven R. Finch.
a(n) ~ A086318 * A086317^(n-1) / n^(3/2). - Vaclav Kotesovec, Apr 19 2016

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 39, 89, 211, 507, 1238, 3057, 7639, 19241, 48865, 124906, 321198, 830219, 2156010, 5622109, 14715813, 38649152, 101821927, 269010485, 712566567, 1891993344, 5034704828, 13425117806, 35866550869, 95991365288, 257332864506, 690928354105

Offset: 0

N. J. A. Sloane

Number of unlabeled rooted trees in which each node has out-degree <= 3.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000625 for the analogous sequence with stereoisomers counted.
In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The out-degree is then <= 3.
Other descriptions of this sequence: quartic planted trees with n nodes; ternary rooted trees with n nodes and height at most 3.
The number of aliphatic amino acids with n carbon atoms in the side chain, and no rings or double bonds, has the same growth as this sequence. - Konrad Gruetzmann, Aug 13 2012

			From _Joerg Arndt_, Feb 25 2017: (Start)
The a(5) = 8 rooted trees with 5 nodes and out-degrees <= 3 are:
:         level sequence    out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 4 ]    [ 1 1 1 1 . ]
:  O--o--o--o--o
:
:     2:  [ 0 1 2 3 3 ]    [ 1 1 2 . . ]
:  O--o--o--o
:        .--o
:
:     3:  [ 0 1 2 3 2 ]    [ 1 2 1 . . ]
:  O--o--o--o
:     .--o
:
:     4:  [ 0 1 2 3 1 ]    [ 2 1 1 . . ]
:  O--o--o--o
:  .--o
:
:     5:  [ 0 1 2 2 2 ]    [ 1 3 . . . ]
:  O--o--o
:     .--o
:     .--o
:
:     6:  [ 0 1 2 2 1 ]    [ 2 2 . . . ]
:  O--o--o
:     .--o
:  .--o
:
:     7:  [ 0 1 2 1 2 ]    [ 2 1 . 1 . ]
:  O--o--o
:  .--o--o
:
:     8:  [ 0 1 2 1 1 ]    [ 3 1 . . . ]
:  O--o--o
:  .--o
:  .--o
(End)

N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 62 (quoting Cayley, who is wrong).
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong).
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.397.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.
Handbook of Combinatorics, North-Holland '95, p. 1963.
Knop, Mueller, Szymanski and Trinajstich, Computer generation of certain classes of molecules.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920.
G. Polya, Mathematical and Plausible Reasoning, Vol. 1 Prob. 4 pp. 85; 233.
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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (first 200 terms from N. J. A. Sloane)
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678.
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong). (Annotated scanned copy)
Frederic Chyzak, Enumerating alcohols and other classes of chemical molecules.
Maximilian Fichtner, K. Voigt, and S. Schuster, The tip and hidden part of the iceberg: Proteinogenic and non-proteinogenic aliphatic amino acids, Biochimica et Biophysica Acta (BBA)-General, 2016, Volume 1861, Issue 1, Part A, January 2017, pp. 3258-3269.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478.
K. Grützmann, S. Böcker, and S. Schuster, Combinatorics of aliphatic amino acids, Naturwissenschaften, Vol. 98, No. 1, 79-86, 2011.
H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.
H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3045. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly.] (Annotated scanned copy)
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2. (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 20, Eq. (G); p. 27, Eq. 2.1.
Marko Riedel, Chris Grossack, Counting trees with a degree constraint, Math StackExchange.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Wikipedia, Polya's enumeration theorem.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000599, A000600, A000602, A000625, A000628, A000678, A010372, A010373, A086194, A086200, A261340.
Cf. A292553, A292554, A292555, A292556.
Cf. A000081, A001190, A014591, A032305, A295461, A298118, A298120, A298204, A298422, A298426.
Column k=3 of A299038.

N := 45; G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2,G000598)*G000598/2+subs(z=z^3,G000598)/3)+O(z^(N+1)),z,N+1): t[ i ] := G000598: i := i+1: od: A000598 := n->coeff(G000598,z,n);
# Another Maple program for g.f. G000598:
G000598 := 1; f := proc(n) global G000598; coeff(series(1+(1/6)*x*(G000598^3+3*G000598*subs(x=x^2,G000598)+2*subs(x=x^3,G000598)),x, n+1),x,n); end; for n from 1 to 50 do G000598 := series(G000598+f(n)*x^n,x,n+1); od; G000598;
spec := [S, {Z=Atom, S=Union(Z, Prod(Z, Set(S, card=3)))}, unlabeled]: [seq(combstruct[count](spec, size=n), n=0..20)];

m = 45; Clear[f]; f[1, x_] := 1+x; f[n_, x_] := f[n, x] = Expand[1+x*(f[n-1, x]^3/6 + f[n-1, x^2]*f[n-1, x]/2 + f[n-1, x^3]/3)][[1 ;; n]]; Do[f[n, x], {n, 2, m}]; CoefficientList[f[m, x], x]
(* second program (after N. J. A. Sloane): *)
m = 45; gf[] = 0; Do[gf[z] = 1 + z*(gf[z]^3/6 + gf[z^2]*gf[z]/2 + gf[z^3]/3) + O[z]^m // Normal, m]; CoefficientList[gf[z], z]  (* Jean-François Alcover, Sep 23 2014, updated Jan 11 2018 *)
b[0, i_, t_, k_] = 1; m = 3; (* m = maximum children *)
b[n_,i_,t_,k_]:= b[n,i,t,k]= If[i<1,0,
  Sum[Binomial[b[i-1, i-1, k, k] + j-1, j]*
  b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
Join[{1},Table[b[n-1, n-1, m, m], {n, 1, 35}]] (* Robert A. Russell, Dec 27 2022 *)

seq(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g,x,x^2)*g/2 + subst(g,x,x^3)/3) + O(x^n)); Vec(g)} \\ Andrew Howroyd, May 22 2018

def seq(n):
    B = PolynomialRing(QQ, 't', n+1);t = B.gens()
    R. = B[[]]
    T = sum([t[i] * z^i for i in range(1,n+1)]) + O(z^(n+1))
    lhs, rhs = T, 1 + z/6 * (T(z)^3 + 3*T(z)*T(z^2) + 2*T(z^3))
    I = B.ideal([lhs.coefficients()[i] - rhs.coefficients()[i] for i in range(n)])
    return [I.reduce(t[i]) for i in range(1,n+1)]
seq(33) # Chris Grossack, Mar 31 2025

G.f. A(x) satisfies A(x) = 1 + (1/6)*x*(A(x)^3 + 3*A(x)*A(x^2) + 2*A(x^3)).

a(n) ~ c * d^n / n^(3/2), where d = 1/A261340 = 2.8154600331761507465266167782426995425365065396907..., c = 0.517875906458893536993162356992854345458168348098... . - Vaclav Kotesovec, Aug 15 2015

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 4, 6, 1, 0, 1, 1, 1, 2, 4, 8, 11, 1, 0, 1, 1, 1, 2, 4, 9, 17, 23, 1, 0, 1, 1, 1, 2, 4, 9, 19, 39, 46, 1, 0, 1, 1, 1, 2, 4, 9, 20, 45, 89, 98, 1, 0, 1, 1, 1, 2, 4, 9, 20, 47, 106, 211, 207, 1, 0

Offset: 0

Alois P. Heinz, Feb 01 2018

			Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   2,   2,   2,   2,   2,   2,   2,   2,   2, ...
  0, 1,   3,   4,   4,   4,   4,   4,   4,   4,   4, ...
  0, 1,   6,   8,   9,   9,   9,   9,   9,   9,   9, ...
  0, 1,  11,  17,  19,  20,  20,  20,  20,  20,  20, ...
  0, 1,  23,  39,  45,  47,  48,  48,  48,  48,  48, ...
  0, 1,  46,  89, 106, 112, 114, 115, 115, 115, 115, ...
  0, 1,  98, 211, 260, 277, 283, 285, 286, 286, 286, ...
  0, 1, 207, 507, 643, 693, 710, 716, 718, 719, 719, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=1-11 give: A000012, A001190(n+1), A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556.
Main diagonal gives A000081 for n>0.
A(2n,n) gives A299039.
Cf. A244372.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
A:= (n, k)-> `if`(n=0, 1, b(n-1$2, k$2)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[ b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
A[n_, k_] := If[n == 0, 1, b[n - 1, n - 1, k, k]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, i, t, k): return 1 if n==0 else 0 if i<1 else sum([binomial(b(i-1, i-1, k, k)+j-1, j)*b(n-i*j, i-1, t-j, k) for j in range(min(t, n//i)+1)])
def A(n, k): return 1 if n==0 else b(n-1, n-1, k, k)
for d in range(15): print([A(n, d-n) for n in range(d+1)]) # Indranil Ghosh, Mar 02 2018, after Maple code

A(n,k) = Sum_{i=0..k} A244372(n,i) for n>0, A(0,k) = 1.

A036718 Number of rooted trees where each node has at most 4 children.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 19, 45, 106, 260, 643, 1624, 4138, 10683, 27790, 72917, 192548, 511624, 1366424, 3666930, 9881527, 26730495, 72556208, 197562840, 539479354, 1477016717, 4053631757, 11149957667, 30732671572, 84871652538, 234802661446, 650684226827

Offset: 0

N. J. A. Sloane

			From _Joerg Arndt_, Feb 25 2017: (Start)
The a(5) = 9 rooted trees with 5 nodes and out-degrees <= 4 are:
:         level sequence    out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 4 ]    [ 1 1 1 1 . ]
:  O--o--o--o--o
:
:     2:  [ 0 1 2 3 3 ]    [ 1 1 2 . . ]
:  O--o--o--o
:        .--o
:
:     3:  [ 0 1 2 3 2 ]    [ 1 2 1 . . ]
:  O--o--o--o
:     .--o
:
:     4:  [ 0 1 2 3 1 ]    [ 2 1 1 . . ]
:  O--o--o--o
:  .--o
:
:     5:  [ 0 1 2 2 2 ]    [ 1 3 . . . ]
:  O--o--o
:     .--o
:     .--o
:
:     6:  [ 0 1 2 2 1 ]    [ 2 2 . . . ]
:  O--o--o
:     .--o
:  .--o
:
:     7:  [ 0 1 2 1 2 ]    [ 2 1 . 1 . ]
:  O--o--o
:  .--o--o
:
:     8:  [ 0 1 2 1 1 ]    [ 3 1 . . . ]
:  O--o--o
:  .--o
:  .--o
:
:     9:  [ 0 1 1 1 1 ]    [ 4 . . . . ]
:  O--o
:  .--o
:  .--o
:  .--o
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. R. Bremner and H. A. Eigendy, Alternating quaternary algebra structures on irreducible representations of sl_2(C), Lin. Alg. Applic. 433 (2010) 1686-1705.
F. Ruskey, Information on Rooted Trees

Cf. A000081, A036717, A036719, A036720, A036721, A036722, A182378, A244372.
Cf. A292553, A292554, A292555, A292556.
Column k=4 of A299038.

A := 1; f := proc(n) global A; local A2,A3,A4; A2 := subs(x=x^2,A); A3 := subs(x=x^3,A); A4 := subs(x=x^4,A);
coeff(series( 1+x*( (A^4+3*A2^2+8*A*A3+6*A^2*A2+6*A4)/2 ), x, n+1), x,n); end;
for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;

a = 1; f[n_] := Module[{a2, a3, a4}, a2 = a /. x -> x^2; a3 = a /. x -> x^3; a4 = a /. x -> x^4; Coefficient[ Series[ 1 + x*(a^4 + 3*a2^2 + 8*a*a3 + 6*a^2*a2 + 6*a4)/24, {x, 0, n + 1}] // Normal, x, n]]; For[n = 1, n <= 30, n++, a = Series[a + f[n]*x^n, {x, 0, n + 1}] // Normal]; CoefficientList[a, x] (* Jean-François Alcover, Jan 16 2013, after Maple *)
b[0, i_, t_, k_] = 1; m = 4; (* m = maximum children *)
b[n_,i_,t_,k_]:= b[n,i,t,k]= If[i<1,0,
   Sum[Binomial[b[i-1, i-1, k, k] + j-1, j]*
   b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
PrependTo[Table[b[n-1, n-1, m, m], {n, 1, 30}], 1] (* Robert A. Russell, Dec 27 2022 *)

G.f. satisfies A(x) = 1 + x*cycle_index(Sym(4), A(x)).
a(n) = Sum_{j=1..4} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017
a(n) / a(n+1) ~ 0.343520104570489046632074698738792654644751898257681287407149... - Robert A. Russell, Feb 11 2023

Better description from Frank Ruskey, Sep 23 2000

A036722 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(6), A(x)).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 114, 283, 710, 1816, 4690, 12267, 32338, 85978, 230080, 619521, 1676808, 4560286, 12454272, 34143682, 93928091, 259208006, 717375068, 1990625390, 5537142610, 15436744525, 43124847431, 120708508008, 338477040445, 950714584576

Offset: 0

N. J. A. Sloane

nonn

a(n) is also the number of rooted trees where each node has at most 6 children. [Patrick Devlin, Apr 29 2012]

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000081, A036717, A036718, A036719, A036720, A036721, A182378, A244372, A292553, A292554, A292555, A292556.

Column k=6 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 6$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 6, 6]];
Table[a[n] , {n, 0, 35}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

a(n) = Sum_{j=1..6} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017

a(n) / a(n+1) ~ 0.338887196052856714304749078960983936661485522864792573284374... - Robert A. Russell, Feb 11 2023

A036721 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(5), A(x)).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 47, 112, 277, 693, 1766, 4547, 11852, 31146, 82534, 220149, 590834, 1593951, 4320723, 11761394, 32138301, 88121176, 242383729, 668607115, 1849194691, 5126800907, 14245679652, 39666239726, 110661514973, 309280533011, 865839831118

Offset: 0

N. J. A. Sloane

nonn

Also the number of rooted trees where each node has at most 5 children. [Patrick Devlin, Apr 30 2012]

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt, The a(6) = 20 rooted trees with 5 nodes and out-degrees <= 5

Cf. A000081, A036717, A036718, A036719, A036720, A036722, A182378, A244372, A292553, A292554, A292555, A292556.

Column k=5 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 5$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 5, 5]];
Table[a[n], {n, 0, 35}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

a(n) = Sum_{j=1..5} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017

a(n) / a(n+1) ~ 0.340017469151060086823930137816585262710976835711484267209811... - Robert A. Russell, Feb 11 2023

A292556 Number of rooted unlabeled trees on n nodes where each node has at most 11 children.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12485, 32970, 87802, 235355, 634771, 1720940, 4688041, 12824394, 35216524, 97039824, 268238379, 743596131, 2066801045, 5758552717, 16080588286, 44997928902, 126160000878, 354349643101, 996946927831

Offset: 0

Marko Riedel, Sep 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.

Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292554, A292555.

Column k=11 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 11$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[ b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n-1, n-1, 11, 11]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

Functional equation of g.f. is T(z) = z + z*Sum_{q=1..11} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group.
Alternate FEQ is T(z) = 1 + z*Z(S_11)(T(z)).
a(n) = Sum_{j=1..11} A244372(n,j) for n > 0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
Limit_{n->oo} a(n)/a(n+1) = 0.338324339068091181557475416836618315086769320447748735003402... - Robert A. Russell, Feb 11 2023

A292553 Number of rooted unlabeled trees on n nodes where each node has at most 8 children.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 718, 1839, 4757, 12460, 32897, 87592, 234746, 633013, 1715851, 4673320, 12781759, 35093010, 96681705, 267199518, 740580555, 2058042803, 5733101603, 16006590851, 44782679547, 125533577578, 352525803976, 991634575368

Offset: 0

Marko Riedel, Sep 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.
Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 1).
Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 2)
Marko Riedel, Maple code (FEQ 2) optimized for speed.

Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292554, A292555, A292556.

Column k=8 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 8$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 8, 8]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Functional equation of G.f. is T(z) = z + z*Sum_{q=1..8} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is T(z) = 1 + z*Z(S_8)(T(z)).
a(n) = Sum_{j=1..8} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338386042364849957035744926227166370702775721795018600630554... - Robert A. Russell, Feb 11 2023

A292554 Number of rooted unlabeled trees on n nodes where each node has at most 9 children.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1841, 4763, 12477, 32947, 87735, 235162, 634212, 1719325, 4683368, 12810871, 35177357, 96926335, 267909285, 742641309, 2064029034, 5750500663, 16057186086, 44929879114, 125962026154, 353773417487, 995269027339

Offset: 0

Marko Riedel, Sep 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.

Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292555, A292556.

Column k=9 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 9$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 9, 9]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Functional equation of G.f. is T(z) = z + z*Sum_{q=1..9} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is
T(z) = 1 + z*Z(S_9)(T(z)).
a(n) = Sum_{j=1..9} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338343552789108712866488147828528012266693326385052387884853... - Robert A. Russell, Feb 11 2023

A182378 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(7), A(x)).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 285, 716, 1833, 4740, 12410, 32754, 87176, 233547, 629540, 1705809, 4644231, 12697500, 34848694, 95973026, 265142431, 734606478, 2040683413, 5682634446, 15859800889, 44355531103, 124290064228, 348904212741, 981082979409

Offset: 0

Michael Burkhart, Apr 26 2012

nonn

Number of rooted trees where each node has at most 7 children.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000081, A036717, A036718, A036719, A036720, A036721, A036722, A244372, A292553, A292554, A292555, A292556.

Column k=7 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 7$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[ b[i-1, i-1, k, k] + j - 1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n-1, n-1, 7, 7]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 15 2018, after Alois P. Heinz *)

a(n) = Sum_{j=1..7} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017

a(n) / a(n+1) ~ 0.338512011286603947719604869750539045616436718225097926729820... - Robert A. Russell, Feb 11 2023

More terms from Patrick Devlin, Apr 29 2012

cp's OEIS Frontend

A001190 Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A036718 Number of rooted trees where each node has at most 4 children.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A036722 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(6), A(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A036721 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(5), A(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A292556 Number of rooted unlabeled trees on n nodes where each node has at most 11 children.