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A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.
6

%I #30 Jun 01 2024 06:21:06

%S 2,12,52,56,212,216,228,232,240,852,856,868,872,880,916,920,932,936,

%T 944,964,968,976,992,3412,3416,3428,3432,3440,3476,3480,3492,3496,

%U 3504,3524,3528,3536,3552,3668,3672,3684,3688,3696,3732,3736,3748,3752,3760

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

%C 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 From _Ruud H.G. van Tol_, May 13 2024: (Start)

%C Sequence on a lattice:

%C Tree Paths Decimal Count

%C |_ 10 2 1

%C |_._ 1100 12 1

%C |_|_._ 110100 -111000 52,56 2

%C |_|_|_._ 11010100 -11110000 212-240 5

%C |_|_|_|_._ 1101010100-1111100000 852-992 14

%C ... (End)

%H Ruud H.G. van Tol, <a href="/A057547/b057547.txt">Table of n, a(n) for n = 0..2055</a>

%H 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 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 <a href="/index/Ro#RootedTreePlanEncodings">Index entries for encodings of plane rooted trees</a>

%F a(n) = A014486(A057548(n)) and also from n > 0 onward = A079946(A014486(n)).

%F a(n) = alltrees2singletrunked(A014486[n]) (see Maple code below and in A057501).

%p alltrees2singletrunked := n -> pars2binexp([binexp2pars(n)]); # Just surround with extra parentheses.

%o (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 Double-trunked trees: A057517. Cf. also A057548, A057549.

%K nonn,base

%O 0,1

%A _Antti Karttunen_ Sep 07 2000