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A343663
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Number of unlabeled binary rooted semi-identity plane trees with 2*n - 1 nodes.
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3
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1, 1, 2, 4, 12, 34, 108, 344, 1136, 3796, 12920, 44442, 154596, 542336, 1917648, 6825464, 24439008, 87962312, 318087216, 1155090092, 4210494616, 15400782912, 56508464736, 207935588586, 767162495940, 2837260332472, 10516827106016, 39063666532784, 145378611426512
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OFFSET
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1,3
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COMMENTS
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In a semi-identity tree, only the non-leaf branches of any given vertex are required to be distinct. Alternatively, a rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees.
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LINKS
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Samuele Giraudo, Mockingbird lattices, Séminaire Lotharingien de Combinatoire XX, Proceedings of the 34th Conf. on Formal Power, Series and Algebraic Combinatorics (Bangalore, India, 2022).
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FORMULA
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G.f.: x*A(x) where A(x) satisfies A(x) = 1 + x + x*(A(x)^2 - A(x^2)). - Andrew Howroyd, May 07 2021
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EXAMPLE
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The a(1) = 1 through a(5) = 12 trees:
o (oo) ((oo)o) (((oo)o)o) ((((oo)o)o)o)
(o(oo)) ((o(oo))o) (((o(oo))o)o)
(o((oo)o)) (((oo)o)(oo))
(o(o(oo))) ((o((oo)o))o)
((o(o(oo)))o)
((o(oo))(oo))
((oo)((oo)o))
((oo)(o(oo)))
(o(((oo)o)o))
(o((o(oo))o))
(o(o((oo)o)))
(o(o(o(oo))))
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MATHEMATICA
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crsiq[n_]:=Join@@Table[Select[Union[Tuples[crsiq/@ptn]], #=={}||#=={{}, {}}||Length[#]==2&&(UnsameQ@@DeleteCases[#, {}])&], {ptn, Join@@Permutations/@IntegerPartitions[n-1]}];
Table[Length[crsiq[n]], {n, 1, 11, 2}]
(* Second program: *)
m = 29; p[_] = 1;
Do[p[x_] = 1 + x + x (p[x]^2 - p[x^2]) + O[x]^m // Normal, {m}];
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PROG
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(PARI) seq(n)={my(p=O(1)); for(n=1, n, p=1 + x + x*(p^2-subst(p, x, x^2))); Vec(p)} \\ Andrew Howroyd, May 07 2021
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CROSSREFS
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The not necessarily semi-identity version is A000108.
The Matula-Goebel numbers in the non-plane case are A339193.
The not-necessarily binary version is A343937.
A000081 counts unlabeled rooted trees with n nodes.
A331966 counts lone-child avoiding semi-identity trees, ranked by A331965.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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