A206491 Irregular triangle read by rows: T(n,k) is the number of root subtrees with k nodes in the rooted tree having Matula-Goebel number n.

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

Offset

1,8

Comments

A root subtree of a rooted tree G is a subtree of G containing the root.

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 = A061775(n).

Sum of entries in row n = A184160(n).

For the number of all subtrees of a given size, see A212620.

References

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.

Links

Table of n, a(n) for n=1..112.

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

Index entries for sequences related to Matula-Goebel numbers

Example

Row 7 is 1,1,2,1 because the rooted tree with Matula-Goebel number 7 is Y; its five root subtrees have 1, 2, 3, 3, and 4 nodes.

Maple

with(numtheory): V := 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+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: R := proc (n, k) 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 and k = 1 then 1 elif n = 1 and 1 < k then 0 elif bigomega(n) = 1 and k = 1 then 1 elif bigomega(n) = 1 then R(pi(n), k-1) else add(R(r(n), j)*R(s(n), k+1-j), j = 1 .. k) end if end proc: for n to 40 do seq(R(n, k), k = 1 .. V(n)) end do; # yields sequence  in triangular form

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];
R[n_, k_] := Which[n == 1 && k == 1, 1, n == 1 && 1 < k, 0, PrimeOmega[n] == 1 && k == 1, 1, PrimeOmega[n] == 1, R[PrimePi[n], k-1], True, Sum[R[r[n], j]*R[s[n], k+1-j], {j, 1, k}]];
Table[R[n, k], {n, 1, 40}, {k, 1, V[n]}] // Flatten (* Jean-François Alcover, Oct 13 2024, after Emeric Deutsch *)

Crossrefs

Cf. A061775, A184160, A212620

Sequence in context: A146292 A139039 A279061 * A359997 A122172 A030613

Adjacent sequences: A206488 A206489 A206490 * A206492 A206493 A206494

Keyword

nonn,tabf

Author

Emeric Deutsch, May 08 2012

Status

approved