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A212632
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The domination number of the rooted tree with Matula-Goebel number n.
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15
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1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 3, 2, 1, 3, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 1, 4, 3, 2, 2, 3, 2, 3, 3, 3, 3, 1, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 4, 2, 2, 4, 3, 3, 3, 3, 3, 3, 2
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OFFSET
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1,5
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COMMENTS
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The domination number of a simple graph G is the minimum cardinality of a dominating subset of G.
The Matula-Goebel number of a rooted tree can be 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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In A212630 one gives the domination polynomial P(n)=P(n,x) of the rooted tree with Matula-Goebel number n. We have a(n) = least exponent in P(n,x).
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EXAMPLE
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a(5)=2 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C; {A,B} is a dominating subset and there is no dominating subset of smaller cardinality.
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MAPLE
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with(numtheory): P := proc (n) local r, s, A, B, C:
r := n -> op(1, factorset(n)): s := n-> n/r(n):
A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(A(pi(n))+B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc:
B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc:
C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc:
sort(expand(A(n)+B(n))) end proc:
A212632 := n->degree(P(n))-degree(numer(subs(x = 1/x, P(n)))): seq(A212632(n), n = 1 .. 120);
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MATHEMATICA
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A[n_] := Which[n == 1, x, PrimeOmega[n] == 1, x*(A[PrimePi[n]] + B[PrimePi[n]] + c[PrimePi[n]]), True, A[r[n]]*A[s[n]]/x];
B[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, A[PrimePi[n]], True, Expand[B[r[n]]*B[s[n]] + B[r[n]]*c[s[n]] + B[s[n]]*c[r[n]]]];
c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True, Expand[c[r[n]]*c[s[n]]]];
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
P[n_] := Expand[A[n] + B[n]];
a[n_] := Exponent[P[n], x] - Exponent[Numerator[P[n] /. x -> 1/x // Together], x];
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CROSSREFS
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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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