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A001190
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Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n-1 nodes in all).
(Formerly M0790 N0298)
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106
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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 n-node binary rooted trees (every node has out-degree <= 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 out-degree 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).
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
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 307.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 55.
S. 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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.52.
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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, 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, Murray R. Bremner, Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube, arXiv:1903.00813 [math.CO], 2019.
F. Bagherzadeh, M. R. Bremner, S. Madariaga, Jordan trialgebras and post-Jordan algebras, arXiv:1611.01214 [math.RA], 2016.
Nils Berglund, 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.
H. Bottomley, Illustration of initial terms
M. Bremner, S. Madariaga, 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.
Broutin, Nicolas; Flajolet, Philippe The distribution of height and diameter in random non-plane binary trees, Random Struct. Algorithms 41, No. 2, 215-252 (2012).
P. 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, 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, 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, B. N. Cyvin, Enumeration of constitutional isomers of polyenes, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261.
N. G. de Bruijn, 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, 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, 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).
S. R. Finch, Otter's Tree Enumeration Constants [Broken link]
S. R. Finch, Otter's Tree Enumeration Constants [Wayback Machine]
P. Flajolet and R. 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.
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, S. Wagner, The shape of random tanglegrams, arXiv preprint arXiv:1512.01168 [math.CO], 2015.
A Ledda, G Achaz, T Wiehe, L Ferretti, Decomposing the site frequency spectrum: the impact of tree topology on neutrality tests, arXiv preprint arXiv:1510.06748 [q-bio.PE], 2015.
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, 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 out-degrees (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 */
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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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