Cheers,
fourteen new sequences, A079437 - A079444 & A080067 - A080071 & A080090 follow.
%I A079437
%S A079437 1,1,2,3,6,16,36,83,190,448,1056,2514,5872,13806,32424,76609,181434,432062,1032716
%N A079437 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071661.
%C A079437 I.e. number of orbits to which "gatomorphisms" A071661/A071662 partition each A000108(n) Catalan tree structures encoded in A014486[A014137(n-1)..A014138(n-1)].
%R A079437
%H A079437 A. Karttunen, C-program for counting the initial terms of this sequence
%O A079437 0,3
%K A079437 nonn,new
%A A079437 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A079437 Occurs in A073201 as row 13373289. Cf. A079438, A079439, A057507, A079441.
%D A079437
%p A079437
%I A079438
%S A079438 1,1,2,2,2,4,4,4,6,6,6,8,8,8,12,12,12,14,16,16,18,18,22,24,24,24,28,28,28,30,34,34,36,36,38,
%T A079438 40,40,40,46,46,46,48,50,50,52,52,56,58,58,58,62,62,62,64,68,68,70,70,72,74,74,74,80,80,80,
%U A079438 82,84,84,86,86,90,92,92,92,96,96,96,98,102,102,104,104,106,108,108,108,114,114,114,116,118
%N A079438 Number of rooted general plane trees which are symmetric and will stay symmetric also after the underlying plane binary tree has been reflected, i.e. number of integers i in range [A014137(i-1)..A014138(i-1)] such that A057164(i)=i and A057164(A057163(i)) = A057163(i).
%C A079438 Also number of fixpoints in range [A014137(n-1)..A014138(n-1)] of permutation A071661 (= Donaghey's automorphism M "squared"), which is equal to condition A057164(i)=A069787(i)=i, i.e. the size of the intersect of fixpoints of permutations A057164 and A069787 in the same range.
%F A079438 a(0)=a(1)=1, a(n) = 2*(floor((n+1)/3) + (if n>=14) (floor((n-10)/4)+floor((n-14)/8))) [This is the correct formula if the conjecture given in A080070 is true, otherwise it is only a lower bound, although known to be exact for upto very high values of n.]
%H A079438 A. Karttunen, C-program for counting the initial terms of this sequence (empirically)
%H A079438 A. Karttunen, Illustration of the initial terms for trees of sizes n=2..18
%R A079438
%D A079438 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%O A079438 0,3
%K A079438 nonn,new
%A A079438 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A079438 From n>= 2 onward A079440(n) = a(n)/2.
%Y A079438 Occurs in A073202 as row 13373289. Cf. A079437, A079439, A079442, A080070.
%p A079438 A079438 := n -> `if`((n<2),1,2*(floor((n+1)/3) + `if`((n>=14),floor((n-10)/4)+floor((n-14)/8),0)));
%I A079439
%S A079439 1,1,1,3,3,5,12,36,72,147,294,336,1068,5076,5760,14742,58968,135288,328176
%N A079439 The longest cycle in range [A014137(n-1)..A014138(n-1)] of permutation A071661.
%R A079439
%H A079439 A. Karttunen, C-program for counting the initial terms of this sequence
%O A079439 0,4
%K A079439 nonn,new
%A A079439 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A079439 Occurs in A073203 as row 13373289. Cf. A079437, A079438, A079443.
%D A079439
%p A079439
%I A079440
%S A079440 0,0,1,1,1,2,2,2,3,3,3,4,4,4,6,6,6,7,8,8,9,9,11,12,12,12,14,14,14,15,17,17,18,18,19,20,20,20,
%T A079440 23,23,23,24,25,25,26,26,28,29,29,29,31,31,31,32,34,34,35,35,36,37,37,37,40,40,40,41,42,42,
%U A079440 43,43,45,46,46,46,48,48,48,49,51,51,52,52,53,54,54,54,57,57,57,58,59,59,60,60,62,63,63,63
%N A079440 Number of transpositions (2-cycles) in range [A014137(n-1)..A014138(n-1)] of permutation A057505 (= Donaghey's automorphism M).
%D A079440 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%R A079440
%O A079440 0,6
%K A079440 nonn,new
%A A079440 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A079440 From n>= 2 onward a(n) = A079438(n)/2 (with the same reservation). Cf. A079444.
%D A079440
%p A079440 A079440 := n -> floor((n+1)/3) + `if`((n>=14),floor((n-10)/4)+floor((n-14)/8),0);
%I A079441
%S A079441 1,1,1,4,7,24,48,128,259,646,1426,3458,7924,19054,44684,107586,255971,617012,1482096
%N A079441 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071663.
%C A079437 I.e. number of orbits to which "gatomorphisms" A071663/A071664 partition each A000108(n) Catalan tree structures encoded in A014486[A014137(n-1)..A014138(n-1)].
%H A079441 A. Karttunen, C-program for counting the initial terms of this sequence
%O A079441 0,4
%K A079441 nonn,new
%A A079441 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A079441 Occurs in A073201 as row 176609070820803. Cf. A057507, A079437, A079442, A079443.
%D A079441
%p A079441
%I A079442
%S A079442 1,1,0,3,0,9,0,21,0,45,0,99,0,195,0,399,0,801,0
%N A079442 Number of fixpoints in range [A014137(n-1)..A014138(n-1)] of permutation A071661.
%R A079442
%H A079442 A. Karttunen, C-program for counting the initial terms of this sequence
%H A079442 A. Karttunen, Illustration of the initial terms for trees of sizes n=3..11
%D A079442 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%O A079442 0,4
%K A079442 nonn,new
%A A079442 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%D A079442
%Y A079442 A079444(n) = A079442(2n+3)/3.
%Y A079442 Occurs in A073202 as row 176609070820803. Cf. A079438, A079441, A079443.
%p A079442
%I A079443
%S A079443 1,1,2,2,2,5,20,24,48,49,196,224,712,3384,3840,9828,39312,90192,218784
%N A079443 The longest cycle in range [A014137(n-1)..A014138(n-1)] of permutation A071663.
%R A079443
%H A079443 A. Karttunen, C-program for counting the initial terms of this sequence
%O A079443 0,3
%K A079443 nonn,new
%A A079443 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A079443 Occurs in A073203 as row 176609070820803. Cf. A079439, A079441, A079442.
%D A079443
%p A079443
%I A079444
%S A079444 1,3,7,15,33,65,133,267
%N A079444 Number of 3-cycles in range [A014137(2n+2)..A014138(2n+2)] of permutation A057505 (= Donaghey's automorphism M).
%R A079444
%H A079444 A. Karttunen, Illustration of the initial terms for trees of sizes 3..11
%D A079444 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%O A079444 0,2
%K A079444 nonn,new
%A A079444 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%D A079444
%Y A079444 a(n) = A079442(2n+3)/3. Cf. A079440.
%p A079444
%I A080067
%S A080067 1,2,5,4,13,11,12,10,9,36,33,34,29,28,35,30,32,27,25,31,26,24,23,106,102,103,94,93,104,95,97,
%T A080067 83,81,96,82,80,79,105,98,99,85,84,101,89,92,78,75,90,76,71,70,100,86,91,77,72,88,74,69,67,
%U A080067 87,73,68,66,65,328,323,324,310,309,325,311,313,285,283,312,284,282,281,326,314,315,287,286
%N A080067 a(n) = A057163(A057548(A057164(n)))
%R A080067
%O A080067 0,2
%K A080067 nonn,new
%A A080067 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A080067 Iterates starting from zero: A080068. Cf. A080070.
%D A080067
%p A080067
%I A080068
%S A080068 0,1,2,5,11,34,82,287,923,3016,8664,37407,102983,414050,1488140
%N A080068 Iterates of A080067.
%F A080068 a(0) = 0, a(n) = A080067(a(n-1))
%R A080068
%O A080068 0,3
%K A080068 nonn,new
%A A080068 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A080068 Corresponding totally balanced binary sequences in A063171: A080070.
%D A080068
%p A080068
%I A080069
%S A080069 0,2,10,44,178,740,2868,11852,47522,190104,735842,3090116,11777124,48557252,194656036
%N A080069 a(n) = A014486(A080068(n))
%R A080069
%O A080069 0,2
%K A080069 nonn,new
%A A080069 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A080069 Same sequence in binary: A080070.
%D A080069
%p A080069
%I A080070
%S A080070 0,10,1010,101100,10110010,1011100100,101100110100,10111001001100,1011100110100010,101110011010011000,
%T A080070 10110011101001100010,1011110010011011000100,101100111011010001100100,10111001001110110011000100,
%U A080070 1011100110100011011100100100
%N A080070 Encoding of parenthesizations produced by simple iteration starting from empty parentheses, and where each successive parenthesization is obtained from the previous by reflecting it as a general tree/parenthesization, then adding an extra stem beneath the root and then reflecting the underlying binary tree.
%F A080070 a(n) = A007088(A080069(n)) = A063171(A080068(n))
%C A080070 Corresponding Lisp/Scheme S-expressions are (),(()),(()()),(()(())),(()(())()),(()((())())),(()(())(()())),...
%C A080070 Conjecture: only the terms in positions 0,1,2 and 4 are symmetric, i.e. A057164(A080068(n)) = A080068(n) (equivalently: A036044(A080069(n)) = A080069(n)) only when n is one of {0,1,2,4}. If this is true, then the formula given in A079438 is exact. I (AK) have checked this upto n=404631 with no other occurrence of a symmetric (general) tree.
%e A080070 This demonstrates how to get the fourth term 10110010 from the 3rd term 101100. The corresponding binary and general trees plus parenthesizations are shown. The first operation reflects the general tree, the second adds a new stem under the root, and the third reflects the underlying binary tree, which induces changes on the corresponding general tree:
%e A080070 ..............................................
%e A080070 .....\/................\/\/..........\/\/.....
%e A080070 ......\/......\/\/......\/............\/......
%e A080070 .....\/........\/........\/..........\/.......
%e A080070 ......(A057164).(A057548)..(A057163)..........
%e A080070 ........................o.....................
%e A080070 ........................|.....................
%e A080070 ........o.....o.........o...o.........o.......
%e A080070 ........|.....|..........\./..........|.......
%e A080070 ....o...o.....o...o.......o.........o.o.o.....
%e A080070 .....\./.......\./........|..........\|/......
%e A080070 ......@.........@.........@...........@.......
%e A080070 ......2.........3.........4...........5.......
%e A080070 ..[()(())]..[(())()]..[((())())]..[()(())()]..
%e A080070 ...101100....110010....11100100....10110010...
%R A080070
%O A080070 0,2
%K A080070 nonn,new
%A A080070 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A080070 Cf. A079438, A080067, A080071, A057119.
%D A080070
%p A080070
%I A080071
%S A080071 0,1,2,2,3,2,3,3,3,2,4,2,3,3,3,3,4,3,2,3,4,4,3,3,2,4,2,3,4,3,2,4,3,3,3,4,3,2,3,4,
%T A080071 3,2,3,6,3,3,3,4,3,3,3,3,3,2,4,3,2,4,5,3,2,4,3,2,3,3,4,2,4,3,2,3,5,2,4,2,6,3,3,3,
%U A080071 3,3,2,4,2,3,4,5,5,3,3,2,5,2,5,2,3,4,2,4,3,3,4,3,2,4,5,4,4,2,5,2,3,3,4,2,5,4,4,3
%N A080071 The top-level length of each parenthesization/root degree of general trees encoded in A080070.
%F A080071 a(n) = A057515(A080068(n))
%R A080071
%O A080071 0,3
%K A080071 nonn,new
%A A080071 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%Y A080071 Cf. A080070, A080090.
%D A080071
%p A080071
%I A080090
%S A080090 0,1,2,4,10,58,43,191,246,320,2000,1602,4172,7598,21843,36520,27737
%N A080090 a(n) = The first occurrence of n in A080071, 0 for those n>0 which never occur in A080071, and -1 if we don't yet know whether it occurs or not.
%C A080090 17 does not occur among the first 404631 terms of A080071.
%R A080090
%O A080090 0,3
%K A080090 nonn,new
%A A080090 Antti Karttunen (Firstname.Surname@iki.fi) Jan 27 2003
%D A080090 Cf. A080071.
%p A080090
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Yours,
Antti