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A386611 a(n) = Sum_{k=0..n-1} binomial(4*k,k) * binomial(4*n-4*k,n-k-1).

%I A386611 #19 Jul 29 2025 08:40:14
%S A386611 0,1,12,126,1268,12513,122148,1184364,11432100,109997460,1055891248,
%T A386611 10117633542,96812495820,925334377822,8836315646616,84317468847768,
%U A386611 804064275489924,7663595943744876,73009005101019792,695263276434909976,6618709687608909648,62989317586872238689
%N A386611 a(n) = Sum_{k=0..n-1} binomial(4*k,k) * binomial(4*n-4*k,n-k-1).
%F A386611 G.f.: g^2 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
%F A386611 G.f.: g/((1-g) * (1-4*g)^2) where g*(1-g)^3 = x.
%F A386611 a(n) = Sum_{k=0..n-1} binomial(4*k+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
%F A386611 a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+1,k).
%F A386611 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+1,k).
%o A386611 (PARI) a(n) = sum(k=0, n-1, binomial(4*k, k)*binomial(4*n-4*k, n-k-1));
%Y A386611 Cf. A308523, A386565, A386612.
%Y A386611 Cf. A008549, A386613, A386615.
%K A386611 nonn
%O A386611 0,3
%A A386611 _Seiichi Manyama_, Jul 27 2025