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A001190
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Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).
(Formerly M0790 N0298)
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116
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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
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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.
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.
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, Univ. Southern Calif., 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
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
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FORMULA
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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
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EXAMPLE
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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)
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) {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 */
(PARI) {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 */
(Python)
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
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CROSSREFS
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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.
Sequence in context: A036591 A036592 A036656 * A274937 A199142 A090344
Adjacent sequences: A001187 A001188 A001189 * A001191 A001192 A001193
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KEYWORD
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easy,core,nonn,nice,eigen
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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