%I A127301
%S A127301 1,2,4,3,8,6,6,7,5,16,12,12,14,10,12,9,14,19,13,10,13,17,11,32,24,24,
%T A127301 28,20,24,18,28,38,26,20,26,34,22,24,18,18,21,15,28,21,38,53,37,26,37,
%U A127301 43,29,20,15,26,37,23,34,43,67,41,22,29,41,59,31,64,48,48,56,40,48,36
%N A127301 Matula-Goebel signatures for plane general trees encoded by A014486.
%C A127301 This sequence hides a morphism that converts A000108(n) oriented (plane) 
               rooted general trees encoded in range [A014137(n-1)..A014138(n-1)] 
               of A014486 to A000081(n+1) non-oriented rooted general trees, encoded 
               by their Matula-Goebel numbers. The latter encoding is explained 
               in A061773.
%C A127301 If the signature-permutation of a Catalan automorphism SP satisfies the 
               condition A127301(SP(n)) = A127301(n) for all n, then it preserves 
               the non-oriented form of a general tree, which implies also that 
               it is Lukasiewicz-word permuting. Examples of such automorphisms 
               include A072796, A057508, A057509/A057510, A057511/A057512, A057164, 
               A127285/A127286 and A127287/A127288.
%e A127301 A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. 
               As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), 
               encodes the following plane tree:
%e A127301 .....o
%e A127301 .....|
%e A127301 .o...o
%e A127301 ..\./.
%e A127301 ...*..
%e A127301 Matula-Goebel encoding for this tree gives a code number A000040(1) * 
               A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
%e A127301 Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the 
               plane tree:
%e A127301 .o
%e A127301 .|
%e A127301 .o...o
%e A127301 ..\./.
%e A127301 ...*..
%e A127301 Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) 
               * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two 
               trees are identical if one ignores their orientation.
%o A127301 (Scheme:) (define (A127301 n) (*A127301 (A014486->parenthesization (A014486 
               n)))) ;; A014486->parenthesization given in A014486.
%o A127301 (define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 
               (*A127301 t)))) 1 s)))
%Y A127301 a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n. 
               a(|A106191(n)|) = A033844(n-1) for all n >= 1. Cf. A127302.
%Y A127301 Sequence in context: A120242 A054427 A048672 this_sequence A122111 A153212 
               A124833
%Y A127301 Adjacent sequences: A127298 A127299 A127300 this_sequence A127302 A127303 
               A127304
%K A127301 nonn
%O A127301 0,2
%A A127301 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 16 2007