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%I #14 Sep 05 2021 04:28:06
%S 1,4,14,14,86,86,49,86,86,886,886,454,886,886,301,301,301,886,886,301,
%T 454,886,886,13766,13766,6418,13766,13766,3986,3986,3986,13766,13766,
%U 3986,6418,13766,13766,3101,3101,1589,3101,3101,1849,1849,3101,13766
%N Matula-Goebel signatures for plane binary trees encoded by A014486.
%C This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
%C If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
%C A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - _Antti Karttunen_, Jan 03 2013
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(n) = A127301(A057123(n)).
%F Can be also computed directly as a fold, see the Scheme-program. - _Antti Karttunen_, Jan 03 2013
%e A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
%e ........\/.....\/.................\/.....\/...
%e .......\/.......\/.....\/.\/.....\/.......\/..
%e ......\/.......\/.......\_/.......\/.......\/.
%e n=.....4........5........6........7........8..
%e The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.
%o (Scheme with _Antti Karttunen_'s IntSeq-library, definition added here Jan 03 2013):
%o (definec (A127302 n) (*A127302 (A014486->parenthesization (A014486 n))))
%o (define (*A127302 s) (fold-right (lambda (t m) (* (A000040 (*A127302 t)) (A000040 m))) 1 s))
%Y Cf A127301, A153835, A153829.
%K nonn
%O 0,2
%A _Antti Karttunen_, Jan 16 2007
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