Edits of A034731, A057513 & A057546.

%I A034731
%S A034731 1,2,3,7,15,46,133,436,1433,4878,16797,58837,208013,743034,2674457,
%T A034731 9695281,35357671,129646266,477638701,1767268073,6564120555,
%U A034731 24466283818,91482563641,343059672916,1289904147339,4861946609466
%N A034731 Dirichlet convolution of b_n=1 with Catalan numbers.
%C A034731 Also number of objects fixed by permutations A057509/A057510 (induced by shallow rotation of general parenthesizations/plane trees).
%F A034731 a(n) = Sum_{d divides n} C(d-1) where C() are the Catalan numbers (A000108).
%Y A034731 Occurs for first time in A073202 as row 16. Cf. A057546, A000108.
%K A034731 nonn
%O A034731 1,2
%A A034731 Erich Friedman (erich.friedman@stetson.edu). More comments from Antti Karttunen (my_firstname.my_surname@iki.fi) Jan 03 2003.


%I A057513
%S A057513 1,1,2,4,9,21,56,153,451,1357,4212,13308,42898
%N A057513 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.
%F A057513 a(0)=1, a(n) = (1/A003418(n-1))*Sum_{i=1..A003418(n-1)} A079216(n,i) [Needs improvement!]
%H A057513 A. Karttunen, <A HREF="http://www.iki.fi/~kartturi/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(With the complete Scheme source)</I>
%H A057513 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%p A057513 A057513 := proc(n) local i; `if`((0=n),1,(1/A003418(n-1))*add(A079216bi(n,i),i=1..A003418(n-1))); end;
%p A057513 # Or empirically:
%p A057513 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 A057513 CountCycles given in A057502, for other procedures, follow A057511 and A057501.
%Y A057513 Occurs for first time in A073201 as row 12. Cf. A057546 and also A000081.
%K A057513 nonn,more
%O A057513 0,3
%A A057513 Antti Karttunen (my_firstname.my_surname@iki.fi) Sep 03 2000. The formula, which is absolutely impractical in the present form, added Jan 03 2003.


%I A057546
%S A057546 1,1,2,3,5,6,10,11,18,21,34,35,68,69,137,148,316,317,759,760,1869,1915,4833,4834,12796,12802,
%T A057546 34108,34384,92792,92793,254752,254753,703083,704956,1958210,1958231,5485330,5485331,15427026,
%U A057546 15440591,43618394,43618395,123807695,123807696,352561832,352664217,1007481494,1007481495,2887387009
%N A057546 Number of Catalan objects fixed by the gatomorphism A057511/A057512 (Deep rotation of general parenthesizations/plane trees).
%F A057546 a(0)=1, a(n) = A079216(n,1) = Sum_{d|n} A079216(d-1,n/d)
%C A057546 More than A003238, because there exists also parenthesizations like ((() (())) ((()) ())) and (((()) ()) (() (()))) which are fixed by recursive deep rotation, corresponding to Catalan mountain ranges below:
%C A057546 ...../\..../\............................./\......../\
%C A057546 ../\/__\../__\/\.....and.its."dual"....../__\/\../\/__\
%C A057546 ./______\/______\......................./______\/______\
%C A057546 It's obvious that a(p) = a(p-1)+1 for all primes p.
%Y A057546 The first row of A079216. The leftmost edge of the triangle A079217 and also its row sums shifted by one. Occurs for first time in A073202 as row 12. Cf. A057513, A079223-A079227, A034731, A003238.
%p A057546 with(numtheory,divisors); A057546 := proc(n) local d; if(0=n) then RETURN(1); else RETURN(add(A079216bi(d-1,n/d),d=divisors(n))); fi; end;
%K A057546 nonn,new
%O A057546 0,3
%A A057546 Antti Karttunen (my_firstname.my_surname@iki.fi) Sep 07 2000. The formula added Jan 03 2003.