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Number of inequivalent leaf-colorings of planted achiral trees with n nodes.
6

%I #10 Dec 14 2020 01:37:22

%S 1,1,3,6,13,20,43,58,115,171,323,379,1034,1135,2321,4327,8915,9212,

%T 33939,34429,128414,234017,417721,418976,2931624,5096391,11770830,

%U 20357876,64853630,64858195

%N Number of inequivalent leaf-colorings of planted achiral trees with n nodes.

%C In a planted achiral tree, all branches directly under any given branch are identical.

%e Inequivalent representatives of the a(5) = 13 leaf-colorings:

%e (1111) ((111)) ((1)(1)) (((11))) ((((1))))

%e (1112) ((112)) ((1)(2)) (((12)))

%e (1122) ((123))

%e (1123)

%e (1234)

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o G(v)={my(t=2, p=sv(1)); for(i=1, #v, my(d=v[i]); if(d>1, p=sApplyCI(symGroupCycleIndex(d), d, p, t)); t=t*d+1); p}

%o cycleIndex(n)={my(recurse(r,v)=if(r==1, G(v), sumdiv(r-1, d, self()((r-1)/d, concat(d,v))))); recurse(n,[])}

%o a(n)={StructsByCycleIndex(n, cycleIndex(n), n)} \\ _Andrew Howroyd_, Dec 13 2020

%Y Cf. A000081, A001190, A001678, A003238, A004111, A214577, A290689, A304486.

%Y Cf. A318226, A318227, A318229, A318230, A318231, A318234, A339645.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, Aug 21 2018

%E a(9)-a(30) from _Andrew Howroyd_, Dec 11 2020