

A206491


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


5



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
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OFFSET

1,8


COMMENTS

A root subtree of a rooted tree G is a subtree of G containing the root.
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel 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 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
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 MatulaGoebel numbers


EXAMPLE

Row 7 is 1,1,2,1 because the rooted tree with MatulaGoebel 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), k1) else add(R(r(n), j)*R(s(n), k+1j), 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


CROSSREFS

Cf. A061775, A184160, A212620
Sequence in context: A146292 A139039 A279061 * A122172 A030613 A025910
Adjacent sequences: A206488 A206489 A206490 * A206492 A206493 A206494


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, May 08 2012


STATUS

approved



