

A206488


The determinant of the distance matrix of the rooted tree having Matula number n.


0



0, 1, 4, 4, 12, 12, 12, 12, 32, 32, 32, 32, 32, 32, 80, 32, 32, 80, 32, 80, 80, 80, 80, 80, 192, 80, 192, 80, 80, 192, 80, 80, 192, 80, 192, 192, 80, 80, 192, 192, 80, 192, 80, 192, 448, 192, 192, 192, 192, 448, 192, 192, 80, 448, 448, 192, 192, 192, 80, 448
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OFFSET

1,3


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.


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..60.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

The Maple program d finds recursively the distance matrix (time consuming).


EXAMPLE

a(2)=1 because the rooted tree with Matula number 2 is the oneedge tree with distance matrix [(0,1, (1,0)].


MAPLE

with(numtheory): with(linalg): with(LinearAlgebra): 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: d := proc (n) local r, s, C, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: C := proc (A, B) local c: c := proc (i, j) options operator, arrow: A[1, i]+B[1, j+1] end proc: Matrix(RowDimension(A), RowDimension(B)1, c) end proc: a := proc (i, j) if i = 1 and j = 1 then 0 elif 2 <= i and 2 <= j then d(pi(n))[i1, j1] elif i = 1 then 1+d(pi(n))[1, j1] elif j = 1 then 1+d(pi(n))[i1, 1] else end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), a) else Matrix(blockmatrix(2, 2, [d(r(n)), C(d(r(n)), d(s(n))), Transpose(C(d(r(n)), d(s(n)))), SubMatrix(d(s(n)), 2 .. RowDimension(d(s(n))), 2 .. RowDimension(d(s(n))))])) end if end proc: for n to 60 do dd[n] := d(n) end do: seq(Determinant(dd[n]), n = 1 .. 60);


CROSSREFS

Sequence in context: A269629 A241496 A273412 * A273491 A168398 A281913
Adjacent sequences: A206485 A206486 A206487 * A206489 A206490 A206491


KEYWORD

sign


AUTHOR

Emeric Deutsch, Apr 14 2012


STATUS

approved



