0,5

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

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.

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.

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.

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.

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.

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

G.f.: A(x) = x + (1/2)*(A(x)^2 + A(x^2)).

G.f. A(x)=1-sqrt(1-2x-A(x^2)) satisfies A(x)^2-2*A(x)+2x+A(x^2)=0, A(0)=0. - 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

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);

(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))

(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 */

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

Sequence in context: A036590 A036591 A036592 this_sequence A036656 A090344 A130131

Adjacent sequences: A001187 A001188 A001189 this_sequence A001191 A001192 A001193

easy,core,nonn,nice,eigen

N. J. A. Sloane (njas(AT)research.att.com).