%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).
%C A057546 Greater 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.
%H A057546 <a href="Sindx_Par.html#parens">Index entries for sequences related to 
               parenthesizing</a>
%F A057546 a(0)=1, a(n) = A079216(n, 1) = Sum_{d|n} A079216(d-1, n/d)
%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;
%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.
%Y A057546 Sequence in context: A024560 A000039 A053436 this_sequence A138587 A099350 
               A008627
%Y A057546 Adjacent sequences: A057543 A057544 A057545 this_sequence A057547 A057548 
               A057549
%K A057546 nonn
%O A057546 0,3
%A A057546 Antti Karttunen (my_firstname.my_surname(AT)iki.fi) Sep 07 2000. Formula 
               added Jan 03 2003.