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%I #28 Feb 24 2023 02:30:57
%S 1,14,429,16796,742900,35357670,1767263190,91482563640,4861946401452,
%T 263747951750360,14544636039226909,812944042149730764,
%U 45950804324621742364,2622127042276492108820,150853479205085351660700,8740328711533173390046320,509552245179617138054608572,29869166945772625950142417512
%N Catalan trisection: A000108(3*n+1), n>=0.
%C See the comment under A187357 for the o.g.f.s for the general trisection of a sequence.
%F a(n) = C(3*n+1), n>=0, with C(n) = A000108(n) (Catalan).
%F O.g.f.: (sqrt(2*sqrt(1+4*x^(1/3)+16*x^(2/3))-(1+8*x^(1/3))) - sqrt(1-4*x^(1/3)))/(6*x^(2/3)).
%F From _Ilya Gutkovskiy_, Jan 13 2017: (Start)
%F E.g.f.: 3F3(1/2,5/6,7/6; 1,4/3,5/3; 64*x).
%F a(n) ~ 4^(3*n+1)/(3*sqrt(3*Pi)*n^(3/2)). (End)
%F Sum_{n>=0} a(n)/4^n = 2*sqrt(2*sqrt(3) - 3)/3. - _Amiram Eldar_, Mar 16 2022
%F a(n) = Product_{1 <= i <= j <= 3*n} (3*i + j + 2)/(3*i + j - 1). - _Peter Bala_, Feb 22 2023
%t Table[CatalanNumber[3*n+1], {n, 0, 20}] (* _Amiram Eldar_, Mar 16 2022 *)
%Y Cf. A000108, A024492, A048990, A187357 (C(3*n)), A187359 (C(3*n+2)).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Mar 09 2011
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