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a(n) = 4*binomial(5*n,n)/(5*n-1).
4

%I #24 Aug 24 2020 00:58:34

%S 5,20,130,1020,8855,81900,791120,7887660,80560285,838553320,

%T 8863227100,94871786100,1026317094705,11203116342560,123243929011680,

%U 1364973221797900,15207477517956825,170321264840835900,1916512328325665070,21655893753689280120

%N a(n) = 4*binomial(5*n,n)/(5*n-1).

%C a(n) is the number of lattice paths from (0,0) to (4n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/4 are the path's endpoints.

%F a(n) = 5*A118971(n-1).

%F G.f.: 5*x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.

%t Array[4 Binomial[5 #, #]/(5 # - 1) &, 20] (* _Michael De Vlieger_, Aug 21 2020 *)

%o (PARI) a(n) = {4*binomial(5*n,n)/(5*n-1)} \\ _Andrew Howroyd_, Aug 21 2020

%Y Cf. A002294, A118971, A337291.

%K nonn,easy

%O 1,1

%A _Lucas A. Brown_, Aug 21 2020