

A057546


Number of Catalan objects of size n fixed by Catalan Automorphism A057511/A057512 (deep rotation of general parenthesizations/plane trees).


20



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, 34108, 34384, 92792, 92793, 254752, 254753, 703083, 704956, 1958210, 1958231, 5485330, 5485331, 15427026, 15440591, 43618394, 43618395, 123807695, 123807696, 352561832, 352664217, 1007481494, 1007481495, 2887387009
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OFFSET

0,3


COMMENTS

Greater than A003238 because there exists also parenthesizations like ((() (())) ((()) ())) and (((()) ()) (() (()))) which are fixed by recursive deep rotation, corresponding to Catalan mountain ranges below:
...../\..../\............................./\......../\
../\/__\../__\/\.....and.its."dual"....../__\/\../\/__\
./______\/______\......................./______\/______\
It's obvious that a(p) = a(p1)+1 for all primes p.


LINKS

Table of n, a(n) for n=0..48.
Index entries for sequences related to parenthesizing


FORMULA

a(0)=1, a(n) = A079216(n, 1) = Sum_{dn} A079216(d1, n/d)


MAPLE

with(numtheory, divisors); A057546 := proc(n) local d; if(0=n) then RETURN(1); else RETURN(add(A079216bi(d1, n/d), d=divisors(n))); fi; end;


CROSSREFS

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, A079223A079227, A034731, A003238.
Sequence in context: A024560 A000039 A053436 * A138587 A099350 A191173
Adjacent sequences: A057543 A057544 A057545 * A057547 A057548 A057549


KEYWORD

nonn


AUTHOR

Antti Karttunen Sep 07 2000. Formula added Jan 03 2003.


STATUS

approved



