cp's OEIS Frontend

A385251 a(n) = Sum_{k=0..n-1} binomial(4*k-3,k) * binomial(4*n-4*k,n-k-1).

%I A385251 #13 Jul 29 2025 08:39:22
%S A385251 0,1,9,84,790,7452,70401,665692,6298236,59612556,564393460,5344664400,
%T A385251 50621130078,479513718116,4542730477758,43039907282664,
%U A385251 407809863233592,3864303038901996,36619104142640460,347027703183853552,3288802989845088504,31169274939274755312
%N A385251 a(n) = Sum_{k=0..n-1} binomial(4*k-3,k) * binomial(4*n-4*k,n-k-1).
%F A385251 G.f.: (g-1)/(g * (4-3*g)^2) where g=1+x*g^4.
%F A385251 G.f.: g * (1-g)^2/(1-4*g)^2 where g*(1-g)^3 = x.
%F A385251 a(n) = Sum_{k=0..n-1} binomial(4*k-3+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
%F A385251 a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n-2,k).
%F A385251 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-2,k).
%o A385251 (PARI) a(n) = sum(k=0, n-1, binomial(4*k-3, k)*binomial(4*n-4*k, n-k-1));
%Y A385251 Cf. A078995, A308523, A386565, A386611, A386612.
%K A385251 nonn
%O A385251 0,3
%A A385251 _Seiichi Manyama_, Jul 28 2025