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1, 8, 625, 82461, 2414517826, 465696894874, 19586243923520645, 911881322544255111111, 2344958374133795816706574529598, 540549352213084909964319555106232
A014486-encoding of the branch-reduced binomial-mod-2 binary trees.
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12
2, 50, 14642, 3969842, 267572689202, 69427226972978, 4581045692538239282, 301220569271221714981682, 1295918094920364850246919050705202, 332029112115571675270693117549056818
COMMENTS
These are obtained from the stunted binomial-mod-2 zigzag trees ( A080263) either by extending each leaf to a branch of two leaves, or by branch-reducing every other such tree.
A014486-encoding of the "Moose trees".
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10
2, 52, 14952, 4007632, 268874213792, 68836555442592, 4561331969745081152, 300550070677246403229312, 1294530259719904904564091957759232, 331402554328705507772604330809117952
COMMENTS
Meeussen's observation about the orbits of a composition of two involutions F and R states that if the orbit size of the composition (acting on a particular element of the set) is odd, then it contains an element fixed by the other involution if and only if it contains also an element fixed by the other, on the (almost) opposite side of the cycle. Here those two involutions are A057163 and A057164, their composition is Donaghey's "Map M" A057505 and as the trees A080293/ A080295 are symmetric as binary trees and the cycle sizes A080292 are odd, it follows that these are symmetric as general trees.
CROSSREFS
Same sequence in binary: A080974. A036044(a(n)) = a(n) for all n. The number of edges (as general trees): A080978.
Orbit size of each tree A080293(n) under Donaghey's "Map M" Catalan automorphism.
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1, 3, 9, 9, 81, 81, 81, 27, 1701, 1701, 1701, 1701, 2673, 2673, 891, 891
Orbit size of each tree A080293(n) under Meeussen's bf<->df map on binary trees.
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3
0, 2, 280, 47104, 1552115753, 320620847201, 14010400861700086, 666566814219424468355, 1738670860867061382977091021290, 403468080959285491446589623771973
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