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%I #25 Feb 24 2023 02:31:35
%S 1,21,715,29393,1337220,64822395,3282060210,171529806825,
%T 9183676536076,501121108325684,27767032438524099,1558142747453650631,
%U 88366931393503350700,5056959295818949067010,291650059796498346544020,16934386878595523443214745,989130828878080326811887228,58078935727891217125276922940,3426228463922436748774829232156,202972497563788492865321721683556
%N Catalan trisection: A000108(3*n + 2)/2, n>=0.
%C See the comment under A187357 for the o.g.f.s of the general trisection of a sequence.
%C The sequence C(3*n+2) starts as 2, 42, 1430, 58786, 2674440, 129644790, 6564120420, 343059613650, ...
%F a(n) = C(3*n+2)/2, n>=0, with C(n) = A000108(n).
%F O.g.f.: (3 - sqrt(1 - 4*x^(1/3)) - sqrt(2)*sqrt(sqrt(1 + 4*x^(1/3) + 16*x^(2/3)) +
%F (1 + 2*x^(1/3))))/(12*x).
%F From _Ilya Gutkovskiy_, Jan 21 2017: (Start)
%F E.g.f.: 3F3(5/6,7/6,3/2; 4/3,5/3,2; 64*x).
%F a(n) ~ 8^(2*n+1)/(3*sqrt(3*Pi)*n^(3/2)). (End)
%F Sum_{n>=0} a(n)/4^n = 1 - sqrt(3+2*sqrt(3))/3. - _Amiram Eldar_, Mar 16 2022
%F a(n) = (1/2)*Product_{1 <= i <= j <= 3*n+1} (3*i + j + 2)/(3*i + j - 1). - _Peter Bala_, Feb 22 2023
%t Table[CatalanNumber[3*n+2]/2, {n, 0, 20}] (* _Amiram Eldar_, Mar 16 2022 *)
%Y Cf. A000108, A024492, A048990, A187357 (C(3*n)), A187358 (C(3*n+1)).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Mar 09 2011
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