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A385668 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

%I A385668 #19 Aug 05 2025 11:47:42
%S A385668 1,13,274,6466,160564,4104733,106927384,2822352952,75224906716,
%T A385668 2020064928916,54569506803574,1481263780787122,40369492671395476,
%U A385668 1103922337550185894,30274295947104877312,832318570941153758356,22932288741241396871068,633044952458953424442364
%N A385668 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
%H A385668 Vincenzo Librandi, <a href="/A385668/b385668.txt">Table of n, a(n) for n = 0..350</a>
%F A385668 a(n) = [x^n] (1+x)^(4*n+1)/(1-2*x)^(3*n+1).
%F A385668 a(n) = [x^n] 1/((1-x) * (1-3*x)^(3*n+1)).
%F A385668 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n+1,k).
%F A385668 a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k,k).
%F A385668 a(n) ~ 2^(8*n + 5/2) / (11 * sqrt(Pi*n) * 3^(2*n - 1/2)). - _Vaclav Kotesovec_, Aug 05 2025
%t A385668 Table[Sum[2^(n-k)*Binomial[4*n+1,k]*Binomial[4*n-k,n-k],{k,0,n}],{n,0,35}] (* _Vincenzo Librandi_, Aug 05 2025 *)
%o A385668 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));
%o A385668 (Magma) [&+[2^(n-k) * Binomial(4*n+1,k) * Binomial(4*n-k,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 05 2025
%Y A385668 Cf. A385438.
%K A385668 nonn
%O A385668 0,2
%A A385668 _Seiichi Manyama_, Aug 04 2025