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A050381
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Number of series-reduced planted trees with n leaves of 2 colors.
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19
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2, 3, 10, 40, 170, 785, 3770, 18805, 96180, 502381, 2667034, 14351775, 78096654, 429025553, 2376075922, 13252492311, 74372374366, 419651663108, 2379399524742, 13549601275893, 77460249369658, 444389519874841
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OFFSET
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1,1
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COMMENTS
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Consider the free algebraic system with two commutative associative operators (x+y) and (x*y) and two generators A,B. The number of elements with n occurrences of the generators is 2*a(n) if n>1, and the number of generators if n=1. - Michael Somos, Aug 07 2017
Also the number of semi-lone-child-avoiding rooted trees with n leaves. Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf. For example, the a(1) = 2 through a(3) = 10 trees are:
o (oo) (ooo)
(o) (o(o)) (o(oo))
((o)(o)) (oo(o))
((o)(oo))
(o(o)(o))
(o(o(o)))
((o)(o)(o))
((o)(o(o)))
(o((o)(o)))
((o)((o)(o)))
(End)
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LINKS
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FORMULA
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Doubles (index 2+) under EULER transform.
Product_{k>=1} (1-x^k)^-a(k) = 1 + a(1)*x + Sum_{k>=2} 2*a(k)*x^k. - Michael Somos, Aug 07 2017
a(n) ~ c * d^n / n^(3/2), where d = 6.158893517087396289837838459951206775682824030495453326610366016992093939... and c = 0.1914250508201011360729769525164141605187995730026600722369002... - Vaclav Kotesovec, Aug 17 2018
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EXAMPLE
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For n=2, the 2*a(2) = 6 elements are: A+A, A+B, B+B, A*A, A*B, B*B. - Michael Somos, Aug 07 2017
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MATHEMATICA
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terms = 22;
B[x_] = x O[x]^(terms+1);
A[x_] = 1/(1 - x + B[x])^2;
Do[A[x_] = A[x]/(1 - x^k + B[x])^Coefficient[A[x], x, k] + O[x]^(terms+1) // Normal, {k, 2, terms+1}];
slaurte[n_]:=If[n==1, {o, {o}}, Join@@Table[Union[Sort/@Tuples[slaurte/@ptn]], {ptn, Rest[IntegerPartitions[n]]}]];
Table[Length[slaurte[n]], {n, 10}] (* Gus Wiseman, Feb 07 2020 *)
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PROG
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(PARI) {a(n) = my(A, B); if( n<2, 2*(n>0), B = x * O(x^n); A = 1 / (1 - x + B)^2; for(k=2, n, A /= (1 - x^k + B)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Aug 07 2017 */
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CROSSREFS
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Lone-child-avoiding rooted trees with n leaves are A000669.
Lone-child-avoiding rooted trees with n vertices are A001678.
The locally disjoint case is A331874.
Semi-lone-child-avoiding rooted trees with n vertices are A331934.
Matula-Goebel numbers of these trees are A331935.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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