A212623 -id:A212623 - cp's OEIS frontend

A212618 Sum of the distances between all unordered pairs of vertices of degree 2 in the rooted tree with Matula-Goebel number n.

0, 0, 0, 0, 1, 1, 0, 0, 4, 4, 4, 0, 0, 0, 10, 0, 0, 2, 0, 1, 1, 10, 2, 0, 20, 2, 6, 0, 1, 6, 10, 0, 20, 1, 4, 2, 0, 0, 6, 1, 2, 0, 0, 4, 13, 6, 6, 0, 0, 14, 4, 0, 0, 6, 35, 0, 1, 6, 1, 6, 2, 20, 2, 0, 13, 13, 0, 0, 13, 1, 1, 2, 0, 2, 24, 0, 10, 3, 4, 1, 12, 6, 6, 0, 10, 0, 14, 4, 0, 13, 2, 2, 35, 13

Offset: 1

Emeric Deutsch, May 22 2012

nonn

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.

			a(11)=4 because the rooted tree with Matula-Goebel number 11 is the path tree A - B - C - D - E; the vertices of degree 2 are B, C, and D; we have dist(B,C)+dist(B,D)+dist(C,D) = 1+2+1 = 4.
a(987654321) = 68, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986 [math.CO], 2011-2012.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A206499, A212619, A212620, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

k := 2: with(numtheory): g := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then sort(expand(x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = k then sort(expand(-x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) <> k-1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k-1 then sort(expand(1+g(r(n))+g(s(n)))) elif bigomega(s(n)) = k then sort(expand(-1+g(r(n))+g(s(n)))) else sort(g(r(n))+g(s(n))) end if end proc; with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) = k then a(pi(n))-subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) <> k and bigomega(pi(n)) <> k-1 then a(pi(n)) elif bigomega(s(n)) = k-1 then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))+subs(x = 1, diff(g(r(n)), x))+subs(x = 1, diff(g(s(n)), x)) elif bigomega(s(n)) = k then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))-subs(x = 1, diff(g(r(n)), x))-subs(x = 1, diff(g(s(n)), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);

k = 2;
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 0,
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1,
  Expand[x + x*g[PrimePi[n]]] ,
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k,
  Expand[-x + x*g[PrimePi[n]]],
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k - 1 &&
  PrimeOmega[PrimePi[n]] != k, Expand[x*g[PrimePi[n]]],
  PrimeOmega[s[n]] == k - 1, Expand[1 + g[r[n]] + g[s[n]]],
  PrimeOmega[s[n]] == k, Expand[-1 + g[r[n]] + g[s[n]]], True,
  g[r[n]] + g[s[n]]];
a[n_] := Which[n == 1, 0,
   PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1,
   a[PrimePi[n]] + (D[g[PrimePi[n]], x] /. x -> 1),
   PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k,
   a[PrimePi[n]] - (D[g[PrimePi[n]], x] /. x -> 1),
   PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k &&
    PrimeOmega[PrimePi[n]] != k - 1, a[PrimePi[n]],
   PrimeOmega[s[n]] == k - 1,
   a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) +
    (D[g[r[n]], x] /. x -> 1) + (D[g[s[n]], x] /. x -> 1) ,
    PrimeOmega[s[n]] == k,
   a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) -
    (D[g[r[n]], x] /. x -> 1) - (D[g[s[n]], x] /. x -> 1), True,
   a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) ];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

We give recurrence formulas for the more general case of vertices of degree k (k>=2). Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,k,x) be the generating polynomial of the vertices of degree k of the rooted tree with Matula-Goebel number n with respect to level. We have a(1) = 0; if n = prime(t) and bigomega(t)=k-1 then a(n) = a(t)+[dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t)=k, then a(n) = a(t)-[dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) =/ k and =/ k-1, then a(n)= a(t); if n = r*s with r prime, s>=2, bigomega(s)=k-1, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} +[dg(r)/dx]{x=1} +[dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} - [dg(r)/dx]{x=1} - [dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) != k-1 and != k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}.

A212620 Irregular triangle read by rows: T(n,k) is the number of k-vertex subtrees of the rooted tree with Matula-Goebel number n (n>=1, k>=1).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 3, 1, 4, 3, 3, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 4, 3, 1, 5, 4, 4, 3, 1, 5, 4, 4, 3, 1, 6, 5, 4, 3, 2, 1, 5, 4, 6, 4, 1, 5, 4, 4, 3, 1, 6, 5, 5, 5, 3, 1, 5, 4, 6, 4, 1, 6, 5, 5, 4, 3, 1, 6, 5, 5, 4, 3, 1, 6, 5, 4, 3, 2, 1, 6, 5, 5, 5, 3, 1, 6, 5, 7, 7, 4, 1, 7, 6, 5, 4, 3, 2, 1, 6, 5, 5, 5, 3, 1, 7, 6, 6, 7, 6, 3, 1, 6, 5, 6, 6, 4, 1, 6, 5, 5, 4, 3, 1, 7, 6, 6, 6, 5, 3, 1

Offset: 1

Emeric Deutsch, May 23 2012

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.
Number of entries in row n is A061775(n) (number of vertices).
Sum of entries in row n is A184161(n) (number of subtrees).
For the number of subtrees containing the root, see A206491.

			T(7,2)=3 because the rooted tree with Matula-Goebel number 7 is Y, having 3 subtrees with 2 vertices.
Row 3 is 3,2,1 because the rooted tree with Matula-Goebel number 3 is the path tree  a - b - c, having 3 subtrees with 1 node each (a, b, c), 2 subtrees with 2 nodes each (ab, bc), and 1 subtree with 3 nodes (abc).
Triangle begins:
  1;
  2,1;
  3,2,1;
  3,2,1;
  4,3,2,1;
  4,3,2,1;
  4,3,3,1;
  4,3,3,1;
  5,4,3,2,1;
  5,4,3,2,1;
  5,4,3,2,1;
  5,4,4,3,1;
  ...

I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
R. E. Jamison, Alternating Whitney sums and matching in trees, part 1, Discrete Math., 67, 1987, 177-189.
R. E. Jamison, Alternating Whitney sums and matching in trees, part 2, Discrete Math., 79, 1989/90, 177-189.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers Discrete Math., 150 (1996), 131-142.
Index entries for sequences related to Matula-Goebel numbers

Cf. A061775, A184161, A206491, A212618, A212619, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

with(numtheory):
R := proc (n) local r, s:
  r := proc (n) options operator, arrow: op(1, factorset(n)) end proc:
  s := proc (n) options operator, arrow: n/r(n) end proc:
  if n = 1 then x elif bigomega(n) = 1 then sort(expand(x+x*R(pi(n)))) else sort(expand(R(r(n))*R(s(n))/x)) end if
end proc:
G := proc (n) local r, s:
  r := proc (n) options operator, arrow: op(1, factorset(n)) end proc:
  s := proc (n) options operator, arrow: n/r(n) end proc:
  if n = 1 then x elif bigomega(n) = 1 then sort(expand(R(n)+G(pi(n)))) else sort(G(r(n))+G(s(n))+R(n)-R(r(n))-R(s(n))) end if
end proc:
WH := proc (n) options operator, arrow: seq(coeff(G(n), x, k), k = 1 .. nops(G(n)))
end proc:
for n to 30 do WH(n) end do; # yields sequence in triangular form

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
R[n_] := Which[n == 1, x, PrimeOmega[n] == 1, Expand[x + x*R[PrimePi[n]]], True, Expand[R[r[n]]* R[s[n]]/x]];
G[n_] := Which[n == 1, x, PrimeOmega[n] == 1, Expand[R[n] + G[PrimePi[n]]], True, Expand[G[r[n]] + G[s[n]] + R[n] - R[r[n]] - R[s[n]]]];
WH[n_] := Rest@CoefficientList[G[n], x];
Table[WH[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Jun 19 2024, after Maple code *)

There exist recursion formulas for the generating polynomial G(n)=G(n,x) of the subtrees with respect to the number of vertices. One introduces also the generating polynomial R(n)=R(n,x) of the root subtrees (subtrees containing the root) with respect to the number of vertices. There is a Maple program for R(n) and one for G(n). From G(n) one extracts the entries of the triangle.

A212619 Sum of the distances between all unordered pairs of vertices of degree 3 in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 3, 0

Offset: 1

Emeric Deutsch, May 22 2012

nonn

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.

			a(28)=1 because the rooted tree with Matula-Goebel number 28 is obtained by joining the trees I, I, and Y at  their roots; it has 2 vertices of degree 3 (the root and the center of Y), the distance between them is 1.
a(987654321) = 22, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986, 2011-2012.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A206499, A212618, A212620, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

k := 3: with(numtheory): g := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then sort(expand(x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = k then sort(expand(-x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) <> k-1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k-1 then sort(expand(1+g(r(n))+g(s(n)))) elif bigomega(s(n)) = k then sort(expand(-1+g(r(n))+g(s(n)))) else sort(g(r(n))+g(s(n))) end if end proc; with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) = k then a(pi(n))-subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) <> k and bigomega(pi(n)) <> k-1 then a(pi(n)) elif bigomega(s(n)) = k-1 then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))+subs(x = 1, diff(g(r(n)), x))+subs(x = 1, diff(g(s(n)), x)) elif bigomega(s(n)) = k then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))-subs(x = 1, diff(g(r(n)), x))-subs(x = 1, diff(g(s(n)), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);

k = 3;
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1, Expand[x + x*g[PrimePi[n]]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k, Expand[-x + x*g[PrimePi[n]]], PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k - 1 && PrimeOmega[PrimePi[n]] != k,  Expand[x*g[PrimePi[n]]], PrimeOmega[s[n]] == k - 1, Expand[1 + g[r[n]] + g[s[n]]], PrimeOmega[s[n]] == k, Expand[-1 + g[r[n]] + g[s[n]]], True, g[r[n]] + g[s[n]]];
a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k - 1, a[PrimePi[n]] + (D[g[PrimePi[n]], x] /. x -> 1), PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == k, a[PrimePi[n]] - (D[g[PrimePi[n]], x] /. x -> 1), PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != k && PrimeOmega[PrimePi[n]] != k - 1, a[PrimePi[n]], PrimeOmega[s[n]] == k - 1, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) + (D[g[r[n]], x] /. x -> 1) + (D[g[s[n]], x] /. x -> 1), PrimeOmega[s[n]] == k, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1) - (D[g[r[n]], x] /. x -> 1) - (D[g[s[n]], x] /. x -> 1), True, a[r[n]] + a[s[n]] + (D[g[r[n]]*g[s[n]], x] /. x -> 1)];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

We give recurrence formulas for the more general case of vertices of degree k (k>=2). Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,k,x) be the generating polynomial of the vertices of degree k of the rooted tree with Matula-Goebel number n with respect to level. We have a(1)=0; if n = prime(t) and bigomega(t) = k-1 then a(n) = a(t) +[dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) =k, then a(n) = a(t) - [dg(t)/dx]{x=1}; if n = prime(t) and bigomega(t) != k and != k-1, then a(n) = a(t); if n = r*s with r prime, s>=2, bigomega(s) =k-1, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} +[dg(r)/dx]{x=1} +[dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]{x=1} - [dg(r)/dx]{x=1} - [dg(s)/dx]{x=1}; if n = r*s with r prime, s>=2, bigomega(s) =/ k-1 and =/ k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}.

A212625 Number of vertices in the largest independent vertex subset of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 5, 4, 4, 4, 4, 4, 4, 4, 5, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 4, 4, 5, 5, 4, 4, 5, 4, 4, 5, 6, 4, 4, 4, 5, 4, 5, 5, 5, 4, 4, 5, 5, 5, 4, 4, 6, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 4, 4, 5, 6, 5, 5

Offset: 1

Emeric Deutsch, Jun 01 2012

nonn

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent.

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.

			a(5)=2 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}; their sizes are 0,1,and 2.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212624, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

with(numtheory): A := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: a := proc (n) options operator, arrow: degree(P(n)) end proc: seq(a(n), n = 1 .. 120);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := Which[n == 1, {x, 1}, PrimeOmega[n] == 1, {x*A[PrimePi[n]][[2]], A[PrimePi[n]][[1]] + A[PrimePi[n]][[2]]}, True, {A[r[n]][[1]] * A[s[n]][[1]]/x, A[r[n]][[2]] * A[s[n]][[2]]}];
P[n_] := A[n] // Total;
a[n_] := Exponent[P[n], x];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

In A212623 one finds the generating polynomial P(n,x) with respect to the number of vertices of the independent vertex subsets of the rooted tree with Matula-Goebel number n. We have a(n)=degree(P(n,x)).

A212627 Irregular triangle read by rows: T(n,k) is the number of maximal independent vertex subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=1).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 0, 3, 0, 3, 1, 0, 1, 1, 0, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 4, 1, 0, 0, 1, 0, 1, 2, 0, 0, 5, 1, 0, 0, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 1, 4, 0, 0, 5, 0, 1, 0, 2, 0, 0, 6, 1, 0, 0, 5, 0, 0, 7, 1, 0, 0, 2, 1, 0, 2, 1, 1, 0, 0, 4, 2, 0, 1, 4, 1, 0, 0, 0, 1, 0, 0, 6, 1, 0, 2, 1, 1, 0, 1, 1, 3, 0, 0, 1, 4, 0, 1, 0, 2, 0, 1, 0, 2, 0, 0, 4, 2, 0, 2, 0, 1, 1, 0, 0, 5, 0, 0, 2, 3, 0, 0, 2, 1, 0, 1, 1, 3, 0, 0, 3, 6

Offset: 1

Emeric Deutsch, Jun 08 2012

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. An independent vertex subset S of a tree is said to be maximal if every vertex that is not in S is joined by an edge to at least one vertex of S.
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.
Number of entries in row n is A212625(n).
Sum of entries in row n = A212628(n).
Sum(k*T(n,k), k>=1) = A212629(n).

			Row 11 is 0, 3, 1 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices R - A - B - C - D; the maximal independent vertex subsets are {R,C}, {A,C}, {A,D}, and {R,B,D}, i.e. none of size 1, three of size 2, and  one of size 3.
Triangle starts:
  1;
  2;
  1,2;
  1,1;
  0,3;
  ...

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2001.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
H. S. Wilf, The number of maximal independent sets in a tree, SIAM J. Alg. Disc. Math., 7, 1986, 125-130.
Index entries for sequences related to Matula-Goebel numbers

Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212624, A212625, A212626, A212628, A212629, A212630, A212631, A212632.

with(numtheory): P := proc (n) local r, s, A, B, C: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(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: if n = 1 then x else sort(expand(A(n)+B(n))) end if end proc: for n to 12 do seq(coeff(P(n), x, j), j = 1 .. degree(P(n))) end do; # yields sequence in triangular form

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := Which[n == 1, x, PrimeOmega[n] == 1, x*(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, 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,  c[r[n]]*c[s[n]]];
P[n_] := A[n] + B[n];
T[n_] := Rest@CoefficientList[P[n], x];
Table[T[n], {n, 1, 50}] // Flatten (* Jean-François Alcover, Jun 19 2024, after Maple code *)

Let A(n)=A(n,x), B(n)=B(n,x), C(n)=C(n,x) be the generating polynomial with respect to size of the maximal independent sets that contain the root, the maximal independent sets that do not contain the root, and the independent sets which are not maximal but become maximal if the root is removed, respectively. We have : A(1)=x, B(1)=0, C(1)=1, A(t-th prime) = x[B(t) + C(t)], B(t-th prime) = A(t), C(t-th prime)=B(t), A(rs)=A(r)A(s)/x, B(rs)=B(r)B(s)+B(r)C(s)+B(s)C(r), C(rs)=C(r)C(s) (r,s>=2). The generating polynomial of the maximal independent vertex subsets of the r oo ted tree with Matula-Goebel number n, with respect to number of vertices, is P(n)=P(n,x)=A(n)+B(n). The Maple program is based on these relations.

A212628 Number of maximal independent vertex subsets in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 2, 2, 4, 4, 4, 3, 3, 3, 5, 2, 3, 5, 2, 4, 4, 5, 5, 3, 7, 5, 8, 3, 4, 6, 5, 2, 7, 4, 5, 5, 3, 3, 6, 4, 5, 5, 3, 5, 9, 8, 6, 3, 4, 8, 5, 5, 2, 9, 9, 3, 4, 6, 4, 6, 5, 7, 8, 2, 8, 8, 3, 4, 9, 6, 4, 5, 5, 5, 11, 3, 7, 8, 5, 4, 16, 6, 8, 5, 7, 5, 8, 5, 3, 10, 6, 8, 9, 9, 5, 3, 8, 5, 13, 8, 5, 6, 9, 5, 9, 3, 3, 9, 6, 10, 6, 3, 6, 5, 13, 6, 12, 5, 5, 6

Offset: 1

Emeric Deutsch, Jun 08 2012

nonn

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. An independent vertex subset S of a tree is said to be maximal if  every vertex that is not in S is joined by an edge to at least one vertex of S.
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.
Let A(n)=A(n,x), B(n)=B(n,x), C(n)=C(n,x) be the generating polynomial with respect to size of the maximal independent sets that contain the root, the maximal independent sets that do not contain the root, and the independent sets which are not maximal but become maximal if the root is removed, respectively. We have A(1)=x, B(1)=0, C(1)=1, A(t-th prime) = x[B(t) + C(t)], B(t-th prime) = A(t), C(t-th prime)=B(t), A(rs)=A(r)A(s)/x, B(rs)=B(r)B(s)+B(r)C(s)+B(s)C(r), C(rs)=C(r)C(s) (r,s>=2). The generating polynomial of the maximal independent vertex subsets with respect to size is P(n, x)=A(n,x)+B(n,x). Then a(n) = P(1,n). The Maple program is based on these relations.

			a(11)=4 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices R - A - B - C - D; the maximal independent vertex subsets are {R,C}, {A,C}, {A,D}, and {R,B,D}.

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
H. S. Wilf, The number of maximal independent sets in a tree, SIAM J. Alg. Disc. Math., 7, 1986, 125-130.
Index entries for sequences related to Matula-Goebel numbers

Cf. A212627.

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*(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: if n = 1 then x else sort(expand(A(n)+B(n))) end if end proc: seq(subs(x = 1, P(n)), n = 1 .. 120);
# For a more efficient calculation, the procedure P() could easily be simplified and optimized to yield A212628(n): remove "sort(expand...)" and replace x with 1 in appropriate places. - M. F. Hasler, Jan 06 2013

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := Which[n == 1, x, PrimeOmega[n] == 1, x*(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, 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, c[r[n]]*c[s[n]]];
P[n_] := A[n] + B[n];
a[n_] := CoefficientList[P[n], x] // Total;
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 20 2024, after Maple code *)

a(n) = Sum_{k>=1} A212627(n,k).

A212621 The overall first Zagreb index of the rooted tree having Matula-Goebel number n.

Original entry on oeis.org

0, 2, 10, 10, 28, 28, 36, 36, 60, 60, 60, 80, 80, 80, 110, 112, 80, 158, 112, 146, 146, 110, 158, 222, 182, 158, 294, 196, 146, 266, 110, 320, 182, 146, 238, 414, 222, 222, 266, 370, 158, 354, 196, 238, 472, 294, 266, 594, 312, 424, 238, 354, 320, 744, 280, 494, 370, 266, 146, 660, 414, 182, 624

Offset: 1

Emeric Deutsch, Jun 01 2012

nonn

The overall first Zagreb index of any simple connected graph G is defined as the sum of the first Zagreb indices of all the subgraphs of G. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.

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.

			a(3)=10 because the rooted tree with Matula-Goebel number 3 is the path tree with 3 vertices R - A - B; the subtrees are R, A, B, RA, AB, and RAB with first Zagreb indices 0, 0, 0, 2, 2, and 6, respectively.

D. Bonchev and N. Trinajstic, Overall molecular descriptors. 3. Overall Zagreb indices, SAR and QSAR in Environmental Research, 12, 2001, 213-236.
F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to Matula-Goebel numbers

Cf. A198339, A196053, A212618, A212619, A212620, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

with(numtheory); Z1 := proc (n) local r, s; r := proc (n) options operator, arrow; op(1, factorset(n)) end proc; s := proc (n) options operator, arrow; n/r(n) end proc; if n = 1 then 0 elif bigomega(n) = 1 then Z1(pi(n))+2+2*bigomega(pi(n)) else Z1(r(n))+Z1(s(n))-bigomega(r(n))^2-bigomega(s(n))^2+bigomega(n)^2 end if end proc; m2union := proc (x, y) sort([op(x), op(y)]) end proc; with(numtheory); MRST := proc (n) local r, s; r := proc (n) options operator, arrow; op(1, factorset(n)) end proc; s := proc (n) options operator, arrow; n/r(n) end proc; if n = 1 then [1] elif bigomega(n) = 1 then [1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))] else [seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))] end if end proc; MNRST := proc (n) local r, s; r := proc (n) options operator, arrow; op(1, factorset(n)) end proc; s := proc (n) options operator, arrow; n/r(n) end proc; if n = 1 then [] elif bigomega(n) = 1 then m2union(mrst[pi(n)], mnrst[pi(n)]) else m2union(mnrst[r(n)], mnrst[s(n)]) end if end proc; MST := proc (n) m2union(mrst[n], mnrst[n]) end proc; for n to 2000 do mrst[n] := MRST(n); mnrst[n] := MNRST(n); mst[n] := MST(n) end do; OZ1 := proc (n) options operator, arrow; add(Z1(MST(n)[j]), j = 1 .. nops(MST(n))) end proc; seq(OZ1(n), n = 1 .. 120); # MRST considers the subtrees that contain the root; MNRST considers the subtrees that do not contain the root; MST considers all subtrees.

A198339(n) gives the sequence of the Matula-Goebel numbers of all the subtrees of the rooted tree with Matula-Goebel number n. A196053(k) is the first Zagreb index of the rooted tree with Matula-Goebel number k.

A212622 The overall second Zagreb index of the rooted tree having Matula-Goebel number n.

Original entry on oeis.org

0, 1, 6, 6, 19, 19, 24, 24, 44, 44, 44, 59, 59, 59, 85, 80, 59, 125, 80, 114, 114, 85, 125, 173, 146, 125, 246, 156, 114, 219, 85, 240, 146, 114, 193, 344, 173, 173, 219, 302, 125, 297, 156, 193, 407, 246, 219, 481, 256, 360, 193, 297, 240, 651, 231, 414, 302, 219, 114, 567, 344, 146, 548, 672, 345, 345, 173, 256, 407, 482, 302, 914, 297

Offset: 1

Emeric Deutsch, Jun 01 2012

nonn

The overall second Zagreb index of any simple connected graph G is defined as the sum of the second Zagreb indices of all the subgraphs of G. The second Zagreb index of a simple connected graph G is the sum of the degree products d(i)d(j) over all edges ij 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.

			a(3)=6 because the rooted tree with Matula-Goebel number 3 is the path tree with 3 vertices R - A - B ; the subtrees are R, A, B, RA, AB, and RAB with second Zagreb indices 0, 0, 0, 1, 1, and 4, respectively.

D. Bonchev and N. Trinajstic, Overall molecular descriptors. 3. Overall Zagreb indices, SAR and QSAR in Environmental Research, 12, 2001, 213-236.
F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to Matula-Goebel numbers

Cf. A098339, A196054, A212618, A212619, A212620, A212621, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

with(numtheory): Z2 := proc (n) local r, s, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: a := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+bigomega(pi(n)) else a(r(n))+a(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then Z2(pi(n))+a(pi(n))+bigomega(pi(n))+1 else Z2(r(n))+Z2(s(n))+a(r(n))*bigomega(s(n))+a(s(n))*bigomega(r(n)) end if end proc: m2union := proc (x, y) sort([op(x), op(y)]) end proc: with(numtheory): MRST := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [1] elif bigomega(n) = 1 then [1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))] else [seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))] end if end proc: MNRST := proc (n) local r, s: r := proc (n) options operator, arrow; op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [] elif bigomega(n) = 1 then m2union(mrst[pi(n)], mnrst[pi(n)]) else m2union(mnrst[r(n)], mnrst[s(n)]) end if end proc: MST := proc (n) m2union(mrst[n], mnrst[n]) end proc: for n to 2000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do: OZ2 := proc (n) options operator, arrow: add(Z2(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OZ2(n), n = 1 .. 120);

A198339(n) gives the sequence of the Matula-Goebel numbers of all the subtrees of the rooted tree with Matula-Goebel number n. A196054(k) is the second Zagreb index of the rooted tree with Matula-Goebel number k.

A212624 Number of vertices in all independent vertex subsets of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 5, 5, 10, 10, 13, 13, 20, 20, 20, 23, 23, 23, 38, 33, 23, 41, 33, 45, 45, 38, 41, 55, 71, 41, 74, 48, 45, 78, 38, 81, 71, 45, 82, 92, 55, 55, 78, 105, 41, 85, 48, 82, 137, 74, 78, 131, 98, 146, 82, 85, 81, 155, 130, 108, 105, 78, 45, 173, 92, 71, 153, 193, 141, 141, 55, 98, 137, 157, 105, 212

Offset: 1

Emeric Deutsch, Jun 01 2012

nonn

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent.
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.
a(n) = Sum_{k>=0} k*A212623(n,k).

			a(5)=10 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}. The total number of vertices is 10.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

with(numtheory): A := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(P(n), x)) end proc: seq(a(n), n = 1 .. 100);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := Which[n == 1, {x, 1}, PrimeOmega[n] == 1, {x*A[PrimePi[n]][[2]], A[PrimePi[n]][[1]] + A[PrimePi[n]][[2]]}, True, {A[r[n]][[1]]*A[s[n]][[1]]/x, A[r[n]][[2]]*A[s[n]][[2]]}];
P[n_] := A[n] // Total;
a[n_] := D[P[n], x] /. x -> 1;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 20 2024, after Maple code *)

In A212623 one finds the generating polynomial P(n,x) with respect to the number of vertices of the independent vertex subsets of the rooted tree with Matula-Goebel number n. We have a(n) = subs(x=1, (d/dx)P(n,x)).

A212626 Number of largest independent vertex subsets of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 2, 5, 1, 1, 1, 4, 5, 2, 1, 5, 1, 1, 1, 2, 4, 1, 1, 1, 3, 4, 2, 2, 2, 1, 5, 3, 1, 3, 6, 1, 2, 2, 1, 1, 3, 3, 1, 9, 5, 1, 1, 2, 1, 2, 4, 1, 1, 1, 7, 7, 2, 1, 6, 1, 1, 4, 3, 4, 2, 1, 1, 8, 3, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 4, 2, 1, 5, 6, 3, 2, 1, 2, 1, 1

Offset: 1

Emeric Deutsch, Jun 01 2012

nonn

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent.

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.

			a(5)= 3 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}; the largest size (namely 2) is attained by 3 of them.

F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
Index entries for sequences related to Matula-Goebel numbers

Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212624, A212625, A212627, A212628, A212629, A212630, A212631, A212632.

with(numtheory): A := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: a := proc (n) options operator, arrow: coeff(P(n), x, degree(P(n))) end proc: seq(a(n), n = 1 .. 120);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := A[n] = Which[n == 1, {x, 1}, PrimeOmega[n] == 1, {x*A[PrimePi[n]][[2]], A[PrimePi[n]][[1]] + A[PrimePi[n]][[2]]}, True, {A[r[n]][[1]]* A[s[n]][[1]]/x, A[r[n]][[2]]*A[s[n]][[2]]}];
P[n_] := A[n] // Total;
a[n_] := Coefficient[P[n], x, Exponent[P[n], x]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

In A212623 one finds the generating polynomial P(n,x) with respect to the number of vertices of the independent vertex subsets of the rooted tree with Matula-Goebel number n. We have a(n) = coefficient of the largest power of x in P(n,x).

cp's OEIS Frontend

A212618 Sum of the distances between all unordered pairs of vertices of degree 2 in the rooted tree with Matula-Goebel number n.

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A212620 Irregular triangle read by rows: T(n,k) is the number of k-vertex subtrees of the rooted tree with Matula-Goebel number n (n>=1, k>=1).

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A212619 Sum of the distances between all unordered pairs of vertices of degree 3 in the rooted tree with Matula-Goebel number n.

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A212625 Number of vertices in the largest independent vertex subset of the rooted tree with Matula-Goebel number n.

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A212627 Irregular triangle read by rows: T(n,k) is the number of maximal independent vertex subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=1).

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A212628 Number of maximal independent vertex subsets in the rooted tree with Matula-Goebel number n.

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A212621 The overall first Zagreb index of the rooted tree having Matula-Goebel number n.

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