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).

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

Author

Emeric Deutsch, May 23 2012

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
  ...
		

References

• 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.

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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.