

A212620


Irregular triangle read by rows: T(n,k) is the number of kvertex subtrees of the rooted tree with MatulaGoebel number n (n>=1, k>=1).


12



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
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OFFSET

1,2


COMMENTS

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 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.


REFERENCES

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.
R. E. Jamison, Alternating Whitney sums and matching in trees, part 1, Discrete Math., 67, 1987, 177189.
R. E. Jamison, Alternating Whitney sums and matching in trees, part 2, Discrete Math., 79, 1989/90, 177189.


LINKS

Table of n, a(n) for n=1..151.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
F. Goebel, On a 11correspondence 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.
Index entries for sequences related to MatulaGoebel numbers


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.


EXAMPLE

T(7,2)=3 because the rooted tree with MatulaGoebel number 7 is Y, having 3 subtrees with 2 vertices.
Row 3 is 3,2,1 because the rooted tree with MatulaGoebel 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).
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;


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


CROSSREFS

Cf. A061775, A184161, A206491, A212618, A212619, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.
Sequence in context: A330727 A174737 A131756 * A194859 A194838 A085014
Adjacent sequences: A212617 A212618 A212619 * A212621 A212622 A212623


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, May 23 2012


STATUS

approved



