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A057547 A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

%I A057547 #30 Jun 01 2024 06:21:06
%S A057547 2,12,52,56,212,216,228,232,240,852,856,868,872,880,916,920,932,936,
%T A057547 944,964,968,976,992,3412,3416,3428,3432,3440,3476,3480,3492,3496,
%U A057547 3504,3524,3528,3536,3552,3668,3672,3684,3688,3696,3732,3736,3748,3752,3760
%N A057547 A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.
%C A057547 This one-to-one correspondence between all rooted plane trees and one node larger, root degree = 1 trees illustrates the fact that INVERT(A000108) = LEFT(A000108). (Catalan numbers shift left under Cameron's A transformation.)
%C A057547 From _Ruud H.G. van Tol_, May 13 2024: (Start)
%C A057547 Sequence on a lattice:
%C A057547 Tree         Paths                  Decimal  Count
%C A057547 |_           10                           2      1
%C A057547 |_._         1100                        12      1
%C A057547 |_|_._       110100    -111000        52,56      2
%C A057547 |_|_|_._     11010100  -11110000    212-240      5
%C A057547 |_|_|_|_._   1101010100-1111100000  852-992     14
%C A057547 ... (End)
%H A057547 Ruud H.G. van Tol, <a href="/A057547/b057547.txt">Table of n, a(n) for n = 0..2055</a>
%H A057547 P. J. Cameron, <a href="http://dx.doi.org/10.1016/0012-365X(89)90081-2">Some sequences of integers</a>, Discrete Math., 75 (1989), 89-102.
%H A057547 P. J. Cameron, <a href="http://dx.doi.org/10.1016/S0167-5060(08)70569-7">Some sequences of integers</a>, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%H A057547 <a href="/index/Ro#RootedTreePlanEncodings">Index entries for encodings of plane rooted trees</a>
%F A057547 a(n) = A014486(A057548(n)) and also from n > 0 onward = A079946(A014486(n)).
%F A057547 a(n) = alltrees2singletrunked(A014486[n]) (see Maple code below and in A057501).
%p A057547 alltrees2singletrunked := n -> pars2binexp([binexp2pars(n)]); # Just surround with extra parentheses.
%o A057547 (PARI) a_rows(N) = my(a=Vec([[2]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, for(i=1, valuation(t, 2), listput(~c, (t<<2)+(2<<i)))); a[r+1]=Vec(c)); a; \\ _Ruud H.G. van Tol_, May 25 2024
%Y A057547 Double-trunked trees: A057517. Cf. also A057548, A057549.
%K A057547 nonn,base
%O A057547 0,1
%A A057547 _Antti Karttunen_ Sep 07 2000