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%I #12 Mar 31 2012 13:21:09
%S 1,1,1,2,1,1,3,2,1,1,5,5,2,1,1,6,11,3,2,1,1,10,26,8,5,2,1,1,11,66,18,
%T 11,3,2,1,1,18,161,43,30,5,5,2,1,1,21,420,104,82,6,14,3,2,1,1,34,1093,
%U 273,233,15,38,5,5,2,1,1,35,2916,702,680,36,111,6,11,3,2,1,1,68,7819,1870
%N Square array A(n>=0,k>=1) (listed antidiagonally: A(0,1)=1, A(1,1)=1, A(0,2)=1, A(2,1)=2, A(1,2)=1, A(0,3)=1, A(3,1)=3, ...) giving the number of n-edge general plane trees fixed by k-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).
%C Note: the counts given here are inclusive, e.g. A(n,6) includes the counts A(n,3) and A(n,2) which in turn both include A(n,1).
%H A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>
%F A(0, k) = 1. A(n, k) = Sum_{r=1..n where r/gcd(r, k) divides n} Sum_{c as each composition of n/(r/gcd(r, k)) into gcd(r, k) parts} Product_{i as each composant of c} A(i-1, lcm(r, k))
%p with(combinat, composition); # composition(n,k) gives ordered partitions of integer n into k parts.
%p [seq(A079216(n),n=0..119)]; A079216 := n -> A079216bi(A025581(n), A002262(n)+1);
%p A079216bi := proc(n,k) option remember; local r; if(0 = n) then RETURN(1); else RETURN(add(PFixedByA057511(n,k,r),r=1..n)); fi; end;
%p PFixedByA057511 := proc(n,k,r) option remember; local ncycles, cyclen, i, c; ncycles := igcd(r,k); cyclen := r/ncycles; if(0 <> (n mod cyclen)) then RETURN(0); else add(mul(A079216bi(i-1,ilcm(r,k)),i=c),c=composition(n/cyclen,ncycles)); fi; end;
%Y A(n, A003418(n)) = A000108(n). The first row: A057546, second: A079223, third: A079224, fourth: A079225, fifth: A079226, sixth: A079227. Cf. also A079217-A079222.
%K nonn,tabl
%O 0,4
%A _Antti Karttunen_ Jan 03 2002
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