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A265102 a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).

%I A265102 #30 Mar 05 2025 05:25:06
%S A265102 1,143,22610,3991995,757398510,150946230006,31170212479588,
%T A265102 6611198199648595,1431806849011462742,315319074704135127010,
%U A265102 70398290295706497441660,15897587681946817926283230,3624898901185998294920196300,833406923656808938891174678092
%N A265102 a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).
%C A265102 Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/4).
%C A265102 Number of maximal faces of the rational associahedron Ass(4*n + 1, 4*n + 5). Number of lattice paths from (0, 0) to (4*n + 5, 4*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (4*n + 1)/(4*n + 5)*x. See Armstrong et al.
%H A265102 Seiichi Manyama, <a href="/A265102/b265102.txt">Table of n, a(n) for n = 0..416</a>
%H A265102 D. Armstrong, B. Rhoades, and N. Williams, <a href="http://arxiv.org/abs/1305.7286">Rational associahedra and noncrossing partitions</a> arxiv:1305.7286v1 [math.CO], 2013.
%F A265102 a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).
%F A265102 (n + 1)*(2*n - 1)*(4*n + 3)*(4*n + 5)*a(n) = 2*(8*n + 1)*(8*n - 1)*(8*n + 3)*(8*n + 5)*a(n-1) with a(0) = 1.
%F A265102 From _Ilya Gutkovskiy_, Feb 28 2017: (Start)
%F A265102 O.g.f.: (4F3(-1/8,1/8,3/8,5/8; -1/2,3/4,5/4; 256*x) - 1)/(2*x).
%F A265102 E.g.f.: 4F4(7/8,9/8,11/8,13/8; 1/2,7/4,2,9/4; 256*x).
%F A265102 a(n) ~ 4^(4*n+1)/(sqrt(Pi)*n^(3/2)). (End)
%p A265102 seq(binomial(8*n + 6, 4*n + 1)/(8*n + 6), n = 0..14);
%t A265102 Table[Binomial[8n+6, 4n+1]/(8n+6), {n, 0, 20}] (* _Vincenzo Librandi_, Dec 09 2015 *)
%o A265102 (PARI) a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6); \\ _Altug Alkan_, Dec 07 2015
%o A265102 (Magma) [Binomial(8*n+6, 4*n+1)/(8*n+6): n in [0..20]]; // _G. C. Greubel_, Feb 16 2019
%o A265102 (Sage) [binomial(8*n+6, 4*n+1)/(8*n+6) for n in (0..20)] # _G. C. Greubel_, Feb 16 2019
%Y A265102 Row 4 of A306444.
%Y A265102 Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265103 (Cat(n + 1/5)).
%K A265102 nonn,easy
%O A265102 0,2
%A A265102 _Peter Bala_, Dec 02 2015