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A057546
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Number of Catalan objects of size n fixed by Catalan Automorphism A057511/A057512 (deep rotation of general parenthesizations/plane trees).
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20
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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
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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MAPLE
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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;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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