%I #30 Jun 24 2024 08:47:34
%S 1,2,6,3,24,8,12,4,20,30,120,10,40,15,72,5,60,24,20,36,36,144,120,12,
%T 252,48,56,18,180,84,720,6,336,72,126,28,60,24,112,42,240,42,90,168,
%U 192,140,504,14,63,288,168,56,30,64,1152,21,56,210,360,96,168,840,96,7,384
%N Product, over all vertices v of the rooted tree with Matula-Goebel number n, of the number of vertices in the subtree with root v.
%C a(n) is called tree factorial. See, for example, the Brouder reference.
%C 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.
%H Ch. Brouder, <a href="https://arxiv.org/abs/hep-th/9904014">Runge-Kutta methods and renormalization</a>, arXiv:hep-th/9904014, 1999; Eur. Phys. J. C 12, 2000, 521-534.
%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H J. Fulman, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r139">Mixing time for a random walk on rooted trees</a>, The Electronic J. of Combinatorics, 16, 2009, R139.
%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H Kevin Ryde, <a href="/A206494/a206494.gp.txt">PARI/GP Code</a>
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F Denote by V(k) the number of vertices of the rooted tree with Matula-Goebel number k. If n is the m-th prime, then a(n) = a(m)*V(n); if n=rs, r,s>=2, then a(n) = a(r)a(s)V(n)/{V(r)V(s)}. The Maple program is based on these recurrence relations.
%e a(7)=12 because the rooted tree with Matula-Goebel number 7 is Y; denoting the vertices in preorder by a,b,c, and d, the number of vertices of the subtrees having these roots are 4, 3, 1, and 1, respectively. a(11)=120 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices; the subtrees have 5,4,3,2,1 vertices.
%p 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: H := 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 V(n)*H(pi(n)) else H(r(n))*H(s(n))*V(n)/(V(r(n))*V(s(n))) end if end proc: seq(H(n), n = 1 .. 100);
%t r[n_] := FactorInteger[n][[1, 1]];
%t s[n_] := n/r[n];
%t V[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + V[PrimePi[n]], True, V[r[n]] + V[s[n]] - 1];
%t H[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, V[n]*H[PrimePi[n]], True, H[r[n]]*H[s[n]]*V[n]/(V[r[n]]*V[s[n]])];
%t Table[H[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 24 2024, after Maple code *)
%o (PARI) \\ See links.
%Y Cf. A196068 (sum of subtree sizes).
%K nonn
%O 1,2
%A _Emeric Deutsch_, May 10 2012