Search: id:A001190
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%I A001190 M0790 N0298
%S A001190 0,1,1,1,2,3,6,11,23,46,98,207,451,983,2179,4850,10905,24631,56011,
%T A001190 127912,293547,676157,1563372,3626149,8436379,19680277,46026618,
%U A001190 107890609,253450711,596572387,1406818759,3323236238,7862958391
%N A001190 Wedderburn-Etherington numbers: binary rooted trees (every node has out-degree 
               0 or 2) with n endpoints (and 2n-1 nodes in all).
%C A001190 Also n-node binary rooted trees (every node has out-degree <= 2) where 
               root has degree 0 or 1.
%C A001190 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).
%C A001190 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
%D A001190 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 55.
%D A001190 S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, 
               J. Molec. Structure (Theochem), 357 (1995), 255-261.
%D A001190 I. M. H. Etherington, Non-associate powers and a functional equation, 
               Math. Gaz. 21 (1937), 36-39 and 153.
%D A001190 I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. 
               Edinburgh, 59 (Part 2, 1938-39), 153-162.
%D A001190 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
%D A001190 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.
%D A001190 F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 
               7 (1984), 191-199.
%D A001190 C. D. Olds, Problem 4277, Amer. Math. Monthly, 56 (1949), 697-699.
%D A001190 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001190 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001190 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 6.52.
%D A001190 J. H. M. Wedderburn, The functional equation g(x^2) = 2ax + [ g(x) ]^2, 
               Ann. Math., 24 (1922-23), 121-140.
%H A001190 T. D. Noe, <a href="b001190.txt">Table of n, a(n) for n = 0..200</a>
%H A001190 H. Bottomley, <a href="a001190.gif">Illustration of initial terms</a>
%H A001190 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001190 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/otter/otter.html">
               Otter's Tree Enumeration Constants</a>
%H A001190 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               72
%H A001190 Piet Hut, <a href="http://www.sns.ias.edu/~piet/">Home Page</a>
%H A001190 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=43">
               Encyclopedia of Combinatorial Structures 43</a>
%H A001190 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=45">
               Encyclopedia of Combinatorial Structures 45</a>
%H A001190 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WeaklyBinaryTree.html">Weakly Binary Tree</a>
%H A001190 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StronglyBinaryTree.html">Strongly Binary Tree</a>
%H A001190 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001190 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A001190 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A001190 <a href="Sindx_Par.html#parens">Index entries for sequences related to 
               parenthesizing</a>
%F A001190 G.f.: A(x) = x + (1/2)*(A(x)^2 + A(x^2)).
%F A001190 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
%F A001190 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.
%F A001190 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
%p A001190 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;
%p A001190 N := 40: G001190 := add(A001190(n)*x^n,n=0..N);
%p A001190 spec := [S,{S=Union(Z,Prod(Z,Set(S,card=2)))},unlabeled]: seq(combstruct[count](spec, 
               size=n), n=0..20);
%o A001190 (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))
%o A001190 (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 */
%Y A001190 Cf. A000108, A001699, A002658, A006894, A003214, A088325.
%Y A001190 Sequence in context: A036590 A036591 A036592 this_sequence A036656 A090344 
               A130131
%Y A001190 Adjacent sequences: A001187 A001188 A001189 this_sequence A001191 A001192 
               A001193
%K A001190 easy,core,nonn,nice,eigen
%O A001190 0,5
%A A001190 N. J. A. Sloane (njas(AT)research.att.com).

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