A212630 -id:A212630 - cp's OEIS frontend

A212623 Irregular triangle read by rows: T(n,k) is the number of independent vertex subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=0).

1, 1, 1, 2, 1, 3, 1, 1, 3, 1, 1, 4, 3, 1, 4, 3, 1, 4, 3, 1, 1, 4, 3, 1, 1, 5, 6, 1, 1, 5, 6, 1, 1, 5, 6, 1, 1, 5, 6, 2, 1, 5, 6, 2, 1, 5, 6, 2, 1, 6, 10, 4, 1, 5, 6, 4, 1, 1, 5, 6, 2, 1, 6, 10, 5, 1, 5, 6, 4, 1, 1, 6, 10, 5, 1, 1, 6, 10, 5, 1, 1, 6, 10, 4, 1, 6, 10, 5, 1, 6, 10, 7, 2, 1, 7, 15

Offset: 1

Emeric Deutsch, Jun 01 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.
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 = A184165(n) = number of independent vertex subset (the Merrifield-Simmons index).
Sum(k*T(n,k), k>=0) = A212624(n) = number of vertices in all independent vertex subsets.
Number of entries in row n = 1 + number of vertices in the largest independent vertex subset = 1 + A212625(n).
Last entry in row n = A212626(n) = number of largest independent vertex subsets.
With the given Maple program, the command P(n) yields the generating polynomial of row n.

			Row 5 is [1,4,3] 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}.
Triangle starts:
  1, 1;
  1, 2;
  1, 3, 1;
  1, 3, 1;
  1, 4, 3;
  ...

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.
H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20, 1982, 16-21.
Index entries for sequences related to Matula-Goebel numbers

Cf. A212618, A212619, A212620, A212621, A212622, A212624, 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: for n to 35 do seq(coeff(P(n), x, k), k = 0 .. 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, 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;
T[n_] := CoefficientList[P[n], x];
Table[T[n], {n, 1, 35}] // Flatten (* Jean-François Alcover, Jun 20 2024, after Maple code *)

Define R(n) =R(n,x) (S(n)=S(n,x)) the generating polynomial of the independent vertex subsets that contain (do not contain) the root of the rooted tree with Matula-Goebel number n. Then R(1)=x, S(1)=1, R(the t-th prime) = x*S(t), S(the t-th prime) = R(t) + S(t); R(rs) = R(r)R(s)/x, S(rs) = S(r)S(s), (r,s>=2).

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 *)

A213669 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the double star graph G(n) obtained by joining with an edge the centers of two star trees each having n+1 vertices (n>=1, k>=2).

Original entry on oeis.org

4, 4, 1, 1, 6, 11, 6, 1, 1, 6, 17, 26, 22, 8, 1, 1, 8, 28, 58, 78, 68, 37, 10, 1, 1, 10, 45, 120, 212, 262, 230, 140, 56, 12, 1, 1, 12, 66, 220, 495, 794, 936, 822, 535, 250, 79, 14, 1, 1, 14, 91, 364, 1001, 2002, 3005, 3446, 3045, 2072, 1071, 406, 106, 16, 1

Offset: 1

Emeric Deutsch, Jul 10 2012

Number of entries in row n is 2n+1.
Sum of entries in row n is (2^n +1)^2 = A028400(n).
The Matula-Goebel number of the rooted tree obtained from G(n), by selecting the center of one of the trees as the root, is 2^n*(2^n-th prime); (knowing this, see A212630 for another approach to find this sequence).
Closely related to the connected domination polynomial of the n-book graph (divided by x^2), which is 1 less in the 3rd-to-last term of each row. - Eric W. Weisstein, May 12 2017

			Row 1 is 4,4,1 because the graph G(1) is the path abcd; there are 4 dominating subsets of size 2 (ac,ad,bc,bd), 4 dominating subsets of size 3 (abc,abd,acd,bcd) and 1 dominating subset of size 4 (abcd).
Triangle starts:
  4,  4,  1;
  1,  6, 11,  6,  1;
  1,  6, 17, 26, 22,  8,  1;
  1,  8, 28, 58, 78, 68, 37, 10,  1;

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251 [math.CO], 2009.
S. Akbari, S. Alikhani, and Y. H. Peng, Characterization of graphs using domination polynomials, European J. Comb., 31, 2010, 1714-1724.
T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Book Graph
Eric Weisstein's World of Mathematics, Connected Dominating Set

Cf. A028400, A212630.

P := proc (n) options operator, arrow: (x^n+x*(1+x)^n)^2 end proc: for n to 9 do seq(coeff(P(n), x, k), k = 2 .. 2*n+2) end do; # yields sequence in triangular form

T[n_, k_] := SeriesCoefficient[(x^n + x (1 + x)^n)^2, {x, 0, k}];
Table[T[n, k], {n, 1, 9}, {k, 2, 2 n + 2}] // Flatten (* Jean-François Alcover, Dec 06 2017 *)

The generating polynomial of row n is (x^n + x(1+x)^n)^2; this is the domination polynomial of the graph G(n).

The domination polynomial of the double star graph obtained by joining with an edge the center of a star tree having m+1 vertices and the center of a star tree having n+1 vertices is (x^m+x(1+x)^m)*(x^n + x(1+x)^n) (m,n >=1).

cp's OEIS Frontend

A212623 Irregular triangle read by rows: T(n,k) is the number of independent vertex subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=0).

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

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

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A213669 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the double star graph G(n) obtained by joining with an edge the centers of two star trees each having n+1 vertices (n>=1, k>=2).

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Examples

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Maple

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