%I #9 Mar 14 2015 14:08:14
%S 1,1,2,4,9,21,56,153,451,1357,4212,13308,42898,140276,465324,1561955,
%T 5300285,18156813,62732842,218405402,765657940
%N Number of separate orbits to which permutations given in A057511/A057512 (induced by deep rotation of general parenthesizations/plane trees) partition each A000108(n) objects encoded by A014486 between (A014138(n-1)+1)-th and (A014138(n))-th terms.
%C It is much faster to compute this sequence empirically with the given C-program than to calculate the terms with the formula in its present form.
%H A. Karttunen, <a href="http://www.iki.fi/~kartturi/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</a> (with the complete Scheme source)
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing empirically the initial terms of this sequence</a>
%F a(0)=1, a(n) = (1/A003418(n-1))*Sum_{i=1..A003418(n-1)} A079216(n, i) [Needs improvement.]
%p A057513 := proc(n) local i; `if`((0=n),1,(1/A003418(n-1))*add(A079216bi(n,i),i=1..A003418(n-1))); end;
%p # Or empirically:
%p DeepRotatePermutationCycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,DeepRotateL(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
%Y CountCycles given in A057502, for other procedures, follow A057511 and A057501.
%Y Similarly generated sequences: A001683, A002995, A003239, A038775, A057507. Cf. also A000081.
%Y Occurs for first time in A073201 as row 12. Cf. A057546 and also A000081.
%K nonn,more
%O 0,3
%A _Antti Karttunen_ Sep 03 2000. The formula, which is absolutely impractical in the present form, added Jan 03 2003.