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A001190 Wedderburn-Etherington numbers: binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n-1 nodes in all).
(Formerly M0790 N0298)
36
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also n-node binary rooted trees (every node has out-degree <= 2) where root has degree 0 or 1.

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

REFERENCES

Cameron, Peter J. 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

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 55.

S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, J. Molec. Structure (Theochem), 357 (1995), 255-261.

de Bruijn, N. G.; Klarner, D. A. 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

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.

Fack, V.; Lievens, S.; Van der Jeugt, J. 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, Mathematical Constants, Cambridge, 2003, pp. 295-316.

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.

A. Gutiérrez-Sánchez, Shen-colored tournaments, thesis, UNAM, 2012.

V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012; http://www.math.sc.edu/~czabarka/JohnsonThesis.pdf.

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.

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.

J. H. M. Wedderburn, The functional equation g(x^2) = 2ax + [g(x)]^2, Ann. Math., 24 (1922-23), 121-140.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

H. Bottomley, Illustration of initial terms

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Filippo Disanto and Thomas Wiehe, Some combinatorial problems on binary rooted trees occurring in population genetics, Arxiv preprint arXiv:1112.1295, 2011

S. R. Finch, Otter's Tree Enumeration Constants

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 72

Piet Hut, Home Page

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 43

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 45

Eric Weisstein's World of Mathematics, Weakly Binary Tree

Eric Weisstein's World of Mathematics, Strongly Binary Tree

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

FORMULA

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

EXAMPLE

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 + ...

MAPLE

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);

MATHEMATICA

n = 32; c[0] = 0; f[x_] = Sum[c[k]*x^k, {k, 0, n}]; eq[0] = Rest[ Thread[ CoefficientList[(-2x + 2f[x] - f[x]^2 - f[x^2])/2, x] == 0]]; s[1] = First[ Solve[ First[eq[0]], c[1]]]; Do[eq[k-1] = Rest[eq[k-2]] /. s[k-1]; s[k] = First[ Solve[ First[eq[k-1]], c[k]]], {k, 2, n}]; Table[c[k], {k, 0, n}] /. Flatten[ Table[s[k], {k, 1, n}]] (* Jean-François Alcover, Jul 22 2011 *)

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 *)

PROG

(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 */

CROSSREFS

Cf. A000108, A001699, A002658, A006894, A006961, A003214, A088325.

Sequence in context: A036591 A036592 * A036656 A199142 A090344 A198662

Adjacent sequences:  A001187 A001188 A001189 * A001191 A001192 A001193

KEYWORD

easy,core,nonn,nice,eigen,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 18 14:05 EDT 2014. Contains 240720 sequences.