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A206487
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Number of ordered trees isomorphic (as rooted trees) to the rooted tree having Matula number n.
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27
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1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 4, 1, 3, 2, 6, 1, 1, 2, 2, 2, 6, 3, 2, 4, 4, 2, 6, 2, 3, 3, 2, 2, 5, 1, 3, 2, 6, 1, 4, 2, 4, 2, 4, 1, 12, 3, 2, 3, 1, 4, 6, 1, 3, 2, 6, 3, 10, 2, 6, 3, 3, 2, 12, 2, 5, 1, 4, 1, 12, 2, 4, 4, 4, 4, 12, 4, 3, 2, 4, 2, 6, 1, 3, 3, 6, 4, 6, 1, 8, 6, 2, 3, 10, 2, 6, 6, 5, 6, 6, 2, 6, 6, 2, 2, 20, 1, 6, 4, 3, 1, 12, 1, 1, 4, 12, 1, 12, 2, 2, 4, 4, 2, 6, 2, 12, 4, 6, 4, 15, 4, 4, 3, 9, 2, 12, 6, 4, 3, 6, 2, 24, 3, 4, 2, 6
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OFFSET
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1,6
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COMMENTS
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The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
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LINKS
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FORMULA
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a(1)=1; denoting by p(t) the t-th prime, if n = p(n_1)^{k_1}...p(n_r)^{k_r}, then a(n) = a(n_1)^{k_1}...a(n_r)^{k_r}*(k_1 + ... + k_r)!/[(k_1)!...(k_r)!] (see Theorem 1 in the Schultz reference, where the exponents k_j of N(n_j) have been inadvertently omitted).
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EXAMPLE
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a(4)=1 because the rooted tree with Matula number 4 is V and there is no other ordered tree isomorphic to V. a(6)=2 because the rooted tree corresponding to n = 6 is obtained by joining the trees A - B and C - D - E at their roots A and C. Interchanging their order, we obtain another ordered tree, isomorphic (as rooted tree) to the first one.
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MAPLE
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with(numtheory): F := proc (n) options operator, arrow: factorset(n) end proc: PD := proc (n) local k, m, j: for k to nops(F(n)) do m[k] := 0: for j while is(n/F(n)[k]^j, integer) = true do m[k] := m[k]+1 end do end do: [seq([F(n)[q], m[q]], q = 1 .. nops(F(n)))] end proc: a := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n)) else mul(a(PD(n)[j][1])^PD(n)[j][2], j = 1 .. nops(F(n)))*factorial(add(PD(n)[k][2], k = 1 .. nops(F(n))))/mul(factorial(PD(n)[k][2]), k = 1 .. nops(F(n))) end if end proc: seq(a(n), n = 1 .. 160);
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MATHEMATICA
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primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]
MGTree[n_Integer]:=If[n===1, {}, MGTree/@primeMS[n]]
treeperms[t_]:=Times@@Cases[t, b:{__}:>Length[Permutations[b]], {0, Infinity}];
Table[treeperms[MGTree[n]], {n, 100}] (* Gus Wiseman, Nov 21 2022 *)
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CROSSREFS
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Positions of 1's are 1 and A214577.
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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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