

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



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).
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;
...


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];


CROSSREFS

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


KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



