A057546 Number of Catalan objects of size n fixed by Catalan Automorphism A057511/A057512 (deep rotation of general parenthesizations/plane trees).

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

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(p-1)+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_{d|n} A079216(d-1, n/d). - Antti Karttunen, Jan 03 2003

Maple

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;

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, A079223-A079227, A034731, A003238.

Sequence in context: A000039 A302600 A053436 * A339514 A138587 A099350

Adjacent sequences: A057543 A057544 A057545 * A057547 A057548 A057549

Keyword

nonn

Author

Antti Karttunen, Sep 07 2000

Status

approved