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A265103 a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).

%I A265103 #27 Mar 05 2025 05:25:02
%S A265103 1,728,482885,347993910,267058714626,214401560777712,
%T A265103 177957899774070416,151516957974714281810,131614194900668669130060,
%U A265103 116186564091895720987588128,103938666796148178180041038716,94020887900502277905668153549928,85855382816448334044679630209920925
%N A265103 a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).
%C A265103 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 + 1/5).
%C A265103 Number of maximal faces of the rational associahedron Ass(5*n + 1, 5*n + 6). Number of lattice paths from (0, 0) to (5*n + 6, 5*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (5*n + 1)/(5*n + 6)*x. See Armstrong et al.
%H A265103 Seiichi Manyama, <a href="/A265103/b265103.txt">Table of n, a(n) for n = 0..333</a>
%H A265103 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 A265103 a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).
%F A265103 (n + 1)*(5*n - 2)*(5*n - 3)*(5*n + 4)*(5*n + 6)*a(n) = 32*(2*n + 1)*(10*n + 1)*(10*n - 1)*(10*n + 3)*(10*n - 3)*a(n-1) with a(0) = 1.
%F A265103 From _Ilya Gutkovskiy_, Feb 28 2017: (Start)
%F A265103 O.g.f.: (5F4(-3/10,-1/10,1/10,3/10,1/2; -3/5,-2/5,4/5,6/5; 1024*x) - 1)/(2*x).
%F A265103 E.g.f.: 5F5(7/10,9/10,11/10,13/10,3/2; 2/5,3/5,9/5,2,11/5; 1024*x).
%F A265103 a(n) ~ 4^(5*n+3)/(5*sqrt(5*Pi)*n^(3/2)). (End)
%p A265103 seq(binomial(10*n + 7, 5*n + 1)/(10*n + 7), n = 0..12);
%t A265103 Table[Binomial[10n+7, 5n+1]/(10n+7), {n, 0, 20}] (* _Vincenzo Librandi_, Dec 09 2015 *)
%o A265103 (PARI)  a(n)=binomial(10*n + 7, 5*n + 1)/(10*n + 7) \\ _Anders Hellström_, Dec 07 2015
%o A265103 (Magma) [Binomial(10*n+7, 5*n+1)/(10*n+7): n in [0..15]]; // _Vincenzo Librandi_, Dec 09 2015
%o A265103 (Sage) [binomial(10*n+7, 5*n+1)/(10*n+7) for n in (0..20)] # _G. C. Greubel_, Feb 16 2019
%Y A265103 Row 5 of A306444.
%Y A265103 Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265102 (Cat(n + 1/4)).
%K A265103 nonn,easy
%O A265103 0,2
%A A265103 _Peter Bala_, Dec 02 2015