0,2

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.

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.

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:

.....o

.....|

.o...o

..\./.

...*..

Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.

Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:

.o

.|

.o...o

..\./.

...*..

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.

(Scheme:) (define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ; ; A014486->parenthesization given in A014486.

(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

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.

Sequence in context: A120242 A054427 A048672 this_sequence A122111 A153212 A124833

Adjacent sequences: A127298 A127299 A127300 this_sequence A127302 A127303 A127304

nonn

Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 16 2007