%I A002658 M1814 N0718
%S A002658 1,1,2,7,56,2212,2595782,3374959180831,5695183504489239067484387,
%T A002658 16217557574922386301420531277071365103168734284282
%N A002658 a(0) = a(1) = 1; for n > 0, a(n+1) = a(n)*(a(0)+...+a(n-1)) + a(n)*(a(n)+1)/
               2.
%C A002658 Number of planted trees in which every node has degree <=3 and of height 
               n; or products of height n when multiplication is commutative but 
               non-associative.
%C A002658 Also called planted 3-trees or planted unary-binary trees.
%C A002658 The next term (which was incorrectly given) is in fact too large to include. 
               See the b-file.
%C A002658 Comment from Marc LeBrun (mlb(AT)well.com): Maximum possible number of 
               distinct new values after applying a commuting operation N times 
               to a single initial value.
%C A002658 Divide the natural numbers in sets of consecutive numbers, starting with 
               {1}, each set with number of elements equal to the sum of elements 
               of the preceding set. The number of elements in the n-th (n>0) set 
               gives a(n). The sets begin {1}, {2}, {3,4}, {5,6,7,8,9,10,11}, ... 
               - Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 16 2002
%D A002658 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002658 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002658 I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. 
               Edinburgh, 59 (Part 2, 1938-39), 153-162.
%D A002658 F. Harary et al., Counting free binary trees..., J. Combin. Inform. System 
               Sciences, 17 (1992), 175-181.
%D A002658 Z. A. Melzak, A note on homogeneous dendrites, Canad. Math. Bull., 11 
               (1968), 85-93.
%H A002658 David Wasserman, <a href="b002658.txt">Table of n, a(n) for n = 0..13</
               a>
%H A002658 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A002658 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A002658 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%p A002658 s := proc(n) local i,j,ans; ans := [ 1 ]; for i to n do ans := [ op(ans),
               ans[ i ]*(add(j,j=ans)-ans[ i ])+ans[ i ]*(ans[ i ]+1)/2 ] od; RETURN(ans); 
               end; t1 := s(10); A002658 := n->t1[n];
%Y A002658 Cf. A006894, A005588. First differences of A072638.
%Y A002658 Sequence in context: A053465 A024027 A079410 this_sequence A034939 A048898 
               A034935
%Y A002658 Adjacent sequences: A002655 A002656 A002657 this_sequence A002659 A002660 
               A002661
%K A002658 nonn,easy,core,nice
%O A002658 0,3
%A A002658 N. J. A. Sloane (njas(AT)research.att.com).
%E A002658 Corrected by David Wasserman, Nov 20 2006