|
%I #16 Jan 11 2024 13:26:32
%S 1,1,2,3,5,6,10,11,18,21,34,35,68,69,137,148,316,317,759,760,1869,
%T 1915,4833,4834,12796,12802,34108,34384,92792,92793,254752,254753,
%U 703083,704956,1958210,1958231,5485330,5485331,15427026,15440591,43618394,43618395,123807695,123807696,352561832,352664217,1007481494,1007481495,2887387009
%N Number of Catalan objects of size n fixed by Catalan Automorphism A057511/A057512 (deep rotation of general parenthesizations/plane trees).
%C Greater than A003238 because there exists also parenthesizations like ((() (())) ((()) ())) and (((()) ()) (() (()))) which are fixed by recursive deep rotation, corresponding to Catalan mountain ranges below:
%C ...../\..../\............................./\......../\
%C ../\/__\../__\/\.....and.its."dual"....../__\/\../\/__\
%C ./______\/______\......................./______\/______\
%C It's obvious that a(p) = a(p-1)+1 for all primes p.
%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>
%F a(0)=1, a(n) = A079216(n, 1) = Sum_{d|n} A079216(d-1, n/d). - _Antti Karttunen_, Jan 03 2003
%p 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 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.
%K nonn
%O 0,3
%A _Antti Karttunen_, Sep 07 2000
|
