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A191516
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Irregular triangle read by rows: a(n,k) is the number of edges with degree k (k>=1) in the rooted tree with Matula-Goebel number n (n>=3).
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1
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2, 2, 2, 1, 2, 1, 0, 3, 0, 3, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 0, 0, 4, 1, 2, 1, 2, 1, 2, 0, 0, 4, 1, 3, 1, 1, 3, 1, 2, 3, 2, 1, 2, 1, 0, 3, 1, 2, 4, 2, 1, 2, 3, 0, 3, 0, 4, 0, 1, 1, 3, 1, 2, 2, 2, 2, 3, 0, 0, 0, 5, 2, 4, 1, 3, 1, 1, 4, 1, 2, 0, 2, 2, 1, 0, 3, 1, 1, 0, 3, 1, 2, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 3, 1, 1
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OFFSET
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3,1
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COMMENTS
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The degree of an edge is the number of edges adjacent to it.
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.
Sum of entries in row n = A196050(n) (= number of edges).
Sum(k*a(n,k), k>=1) = A198332(n) (=sum of edge degrees (the Platt index)).
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REFERENCES
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A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
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LINKS
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FORMULA
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Let f(n)=F(n,x) be the generating polynomial of the edges of the rooted tree with Matula-Goebel number n, with respect to edge-degree. Write f(n)=g(n)+h(n), where g(n) is over the edges emanating from the root and h(n) is over the remaining edges. We have g(1)=0, h(1)=0; if n = prime(t), then g(n)=x^G(t), h(n)=xg(t)+h(t); if n=r*s (r,s>=2), then g(n)=x^G(s)*g(r) + x^G(r)*g(s), h(n)=h(r)+h(s). G(m) denotes the number of prime divisors of m counted with multiplicities.
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EXAMPLE
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Row 5 is 2,1 because the rooted tree with Matula-Goebel number 5 is the path tree ABCD on 4 vertices; AB and CD have degree 1 and BC has degree 2.
Row 7 is 0,3 because the rooted tree with Matula-Goebel number 7 is Y, where no edge has degree 1 and all 3 edges have degree 2.
Triangle starts:
2;
2;
2,1;
2,1;
0,3;
0,3;
2,2;
...
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MAPLE
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with(numtheory): f := proc (n) local r, s, g, h: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: g:=proc(n) if n = 1 then 0 elif bigomega(n) = 1 then x^(bigomega(pi(n))) else x^(bigomega(s(n)))*g(r(n))+x^(bigomega(r(n)))*g(s(n)) fi end: h:=proc(n) if n=1 then 0 elif bigomega(n)=1 then x*g(pi(n))+h(pi(n)) else h(r(n))+h(s(n)) fi end: sort(expand(g(n)+h(n))) end: for n from 3 to 42 do seq(coeff(f(n), x, j), j=1..degree(f(n))) od; # yields sequence in triangular form
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MATHEMATICA
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r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x^(PrimeOmega[PrimePi[n]]), True, x^(PrimeOmega[s[n]])*g[r[n]] + x^(PrimeOmega[r[n]])*g[s[n]]];
h[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*g[PrimePi[n]] + h[PrimePi[n]], True, h[r[n]] + h[s[n]]];
f[n_] := g[n] + h[n];
T[n_] := Rest@CoefficientList[f[n], x];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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