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A386565 a(n) = Sum_{k=0..n-1} binomial(4*k-1,k) * binomial(4*n-4*k,n-k-1).

%I A386565 #24 Jul 29 2025 08:40:18
%S A386565 0,1,11,111,1091,10596,102237,982458,9415539,90063180,860278156,
%T A386565 8208539351,78258171957,745595635084,7099714918062,67574576298276,
%U A386565 642927956583123,6115089154367484,58146652079312580,552769690436583532,5253812277363417836,49925987913040522128
%N A386565 a(n) = Sum_{k=0..n-1} binomial(4*k-1,k) * binomial(4*n-4*k,n-k-1).
%F A386565 G.f.: g*(g-1)/(4-3*g)^2 where g=1+x*g^4.
%F A386565 G.f.: g/(1-4*g)^2 where g*(1-g)^3 = x.
%F A386565 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ).
%F A386565 a(n) = Sum_{k=0..n-1} binomial(4*k-1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
%F A386565 a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n,k).
%F A386565 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k,k).
%e A386565 (1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ) = x + 11*x^2/2 + 37*x^3 + 1091*x^4/4 + 10596*x^5/5 + ...
%o A386565 (PARI) a(n) = sum(k=0, n-1, binomial(4*k-1, k)*binomial(4*n-4*k, n-k-1));
%o A386565 (PARI) my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); concat(0, Vec(g*(g-1)/(4-3*g)^2))
%Y A386565 Cf. A000346, A062236, A386566, A386567.
%Y A386565 Cf. A002293, A006632, A308523.
%K A386565 nonn
%O A386565 0,3
%A A386565 _Seiichi Manyama_, Jul 26 2025