A198339 -id:A198339 - cp's OEIS frontend

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

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.

			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.

A198338 Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 3, 5, 1, 2, 2, 3, 4, 6, 1, 2, 3, 3, 7, 1, 2, 2, 2, 4, 4, 4, 8, 1, 2, 2, 3, 3, 4, 6, 6, 9, 1, 2, 2, 3, 4, 5, 6, 10, 1, 2, 3, 5, 11, 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 8, 12, 1, 2, 3, 3, 4, 6, 6, 7, 15, 1, 2, 2, 3, 3, 4, 5, 6, 6

Offset: 1

Emeric Deutsch, Dec 04 2011

Number of entries in row n is A184160(n). Row n>=2 can be easily identified: it starts with the entry following the first occurrence of n-1 and it ends with the first occurrence of n.

			Row 4 is [1,2,2,4] because the rooted tree with Matula-Goebel number 4 is V and its root subtrees are *, |, |, and V.
Triangle starts:
1;
1,2;
1,2,3;
1,2,2,4;
1,2,3,5;
1,2,2,3,4,6;

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 Review, 10, 1968, 273.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288

Cf. A198339.

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 sort([1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))]) else sort([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: for n to 1000 do mrst[n] := MRST(n) end do;

Row 1 is [1]; if n = p(t) (= the t-th prime), then row n is [1, p(a), p(b), ... ], where [a,b,...] is row t; if n=rs (r,s >=2), then row n consists of the numbers r[i]*s[j], where [r[1], r[2],...] is row r and [s[1], s[2], ...] is row s. The Maple program, based on this recursive procedure, yields row n (<=1000; adjustable) with the command MRST(n).

A198340 The overall Wiener index of the rooted tree having Matula-Goebel number n.

Original entry on oeis.org

0, 1, 6, 6, 21, 21, 24, 24, 56, 56, 56, 67, 67, 67, 126, 80, 67, 161, 80, 154, 154, 126, 161, 197, 252, 161, 354, 188, 154, 333, 126, 240, 252, 154, 311, 440, 197, 197, 333, 414, 161, 411, 188, 311, 683, 354, 333, 545, 384, 636, 311, 411, 240, 921, 462, 510

Offset: 1

Emeric Deutsch, Dec 04 2011

nonn

The overall Wiener index of any connected graph G is defined as the sum of the Wiener indices of all the subgraphs of G. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

			a(4)=6 because the rooted tree with Matula-Goebel number 4 is V; each of the 3 one-vertex subtrees has Wiener index 0, each of the 2 one-edge subtrees has Wiener index 1, and the tree V itself has Wiener index 4; 0+0+0+1+1+4=6.

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 Review, 10, 1968, 273.
D. Bonchev, The overall Wiener index - a new tool for characterization of molecular topology, J. Chem. Inf. Comput. Sci., 2001, 41, 582-592.

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

Cf. A196051, A198339.

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: W := proc (n) local u, v, E, PL: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(u(n))+E(v(n)) end if end proc: PL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n))+PL(pi(n)) else PL(u(n))+PL(v(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then W(pi(n))+PL(pi(n))+1+E(pi(n)) else W(u(n))+W(v(n))+PL(u(n))*E(v(n))+PL(v(n))*E(u(n)) end if end proc: OW := proc (n) options operator, arrow: add(W(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OW(n), n = 1 .. 60);

A198341 The overall hyper-Wiener index of the rooted tree having Matula-Goebel number n.

Original entry on oeis.org

0, 1, 7, 7, 28, 28, 30, 30, 84, 84, 84, 94, 94, 94, 210, 104, 94, 247, 104, 243, 243, 210, 247, 283, 462, 247, 579, 278, 243, 565, 210, 320, 462, 243, 547, 681, 283, 283, 565, 667, 247, 661, 278, 547, 1216, 579, 565, 793, 644, 1174, 547, 661, 320, 1506, 924

Offset: 1

Emeric Deutsch, Dec 04 2011

nonn

The overall hyper-Wiener index of any connected graph G is defined as the sum of the hyper-Wiener indices of all the subgraphs of G. The hyper-Wiener index of a connected graph is (1/2)*Sum [d(i,j)+d(i,j)^2], where d(i,j) is the distance between the vertices i and j and summation is over all unordered pairs of vertices (i,j).
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.
The Maple program yields a(n) by using the command OHW(n) for n<=3000 (adjustable).

			a(4)=7 because the rooted tree with Matula-Goebel number 4 is V; each of the 3 one-vertex subtrees has hyper-Wiener index 0, each of the 2 one-edge subtrees has hyper-Wiener index 1, and the tree V itself has hyper-Wiener index 5; 0+0+0+1+1+5=7.

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 Review, 10, 1968, 273.
D. Bonchev, The overall Wiener index - a new tool for characterization of molecular topology, J. Chem. Inf. Comput. Sci., 2001, 41, 582-592.
X. H. Li and J. J. Lin, The overall hyper-Wiener index, J. Math. Chemistry, 33, 2003, 81-89.

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

Cf. A196060, A198339.

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 3000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do: W := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: HW := proc (n) options operator, arrow: subs(x = 1, diff(W(n), x)+(1/2)*(diff(W(n), `$`(x, 2)))) end proc: OHW := proc (n) options operator, arrow: add(HW(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OHW(n), n = 1 .. 60);

A198339(n) gives the sequence of the Matula-Goebel numbers of all the subtrees of the rooted tree with Matula-Goebel number n. A196060(k) is the hyper-Wiener index of the rooted tree with Matula-Goebel number k.

A184186 Irregular triangle read by rows: row n is the overall Wiener index vector of the rooted tree having Matula-Goebel number n (n>=2).

Original entry on oeis.org

1, 2, 4, 2, 4, 3, 8, 10, 3, 8, 10, 3, 12, 9, 3, 12, 9, 4, 12, 20, 20, 4, 12, 20, 20, 4, 12, 20, 20, 4, 16, 29, 18, 4, 16, 29, 18, 4, 16, 29, 18, 5, 16, 30, 40, 35, 4, 24, 36, 16, 4, 16, 29, 18, 5, 20, 49, 56, 31, 4, 24, 36, 16, 5, 20, 39, 58, 32, 5, 20, 39

Offset: 2

Emeric Deutsch, Dec 04 2011

Component i (i>=1) of the overall Wiener index (number) vector of a graph G is defined as the sum of the Wiener numbers of all i-edge subgraphs of G (see the Bonchev reference, p. 583).
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 = 1st entry in row n = number of edges of the corresponding tree = A196050(n).
Last entry in row n = the Wiener index of the corresponding tree = A196051(n).
Sum of entries in row n = the overall Wiener index of the corresponding tree = A198340(n).
The Maple program yields row n with the command OWV(n) for n<=3000 (adjustable).

			Row n=5 is 3,8,10 because the rooted tree with Matula-Goebel number 5 is the path tree on 4 vertices; each of the three 1-edge subtrees has Wiener index 1, each of the two 2-edge subtrees has Wiener index 4 and the given 3-edge tree itself has Wiener index 10.
Triangle starts (n>=2):
1;
2,4;
2,4;
3,8,10;
3,8,10;
3,12,9;
3,12,9;
4,12,20,20;

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 Review, 10, 1968, 273.
D. Bonchev, The overall Wiener index - a new tool for characterization of molecular topology, J. Chem. Inf. Comput. Sci., 2001, 41, 582-592.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288

Cf. A196050, A196051, A198339, A198340.

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 3000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do: E := 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 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: W := proc (n) local u, v, E, PL: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(u(n))+E(v(n)) end if end proc: PL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n))+PL(pi(n)) else PL(u(n))+PL(v(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then W(pi(n))+PL(pi(n))+1+E(pi(n)) else W(u(n))+W(v(n))+PL(u(n))*E(v(n))+PL(v(n))*E(u(n)) end if end proc: OWV := proc (n) local i, c, g, k: for i from 0 to E(n) do c[i] := 0 end do: g := MST(n): for k to nops(g) do c[E(g[k])] := c[E(g[k])]+W(g[k]) end do: seq(c[i], i = 1 .. E(n)) end proc:

A198339(n) gives the sequence of the Matula-Goebel numbers of all the subtrees of the rooted tree with Matula-Goebel number n. A196051(k) is the Wiener number of the rooted tree with Matula-Goebel number k. A196050(k) is equal to the number of edges of the rooted tree with Matula-Goebel number k. In the Maple program we take the sum of the Wiener indices of all the subtrees, grouped according to number of edges.

cp's OEIS Frontend

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

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

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A198338 Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.

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A198340 The overall Wiener index of the rooted tree having Matula-Goebel number n.

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A198341 The overall hyper-Wiener index of the rooted tree having Matula-Goebel number n.

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A184186 Irregular triangle read by rows: row n is the overall Wiener index vector of the rooted tree having Matula-Goebel number n (n>=2).

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