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A385514 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).

%I A385514 #30 Aug 12 2025 03:10:19
%S A385514 1,7,64,643,6766,73162,805414,8979523,101060326,1145704162,
%T A385514 13064219224,149674343518,1721537039236,19866626222632,
%U A385514 229912254620434,2667252458378083,31009548579437446,361198085246048602,4214267651960927824,49243413868632029338,576179701092650156356
%N A385514 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
%H A385514 Vincenzo Librandi, <a href="/A385514/b385514.txt">Table of n, a(n) for n = 0..350</a>
%F A385514 a(n) = [x^n] (1+x)^(2*n+1)/(1-2*x)^(n+1).
%F A385514 a(n) = [x^n] 1/((1-x) * (1-3*x)^(n+1)).
%F A385514 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k).
%F A385514 a(n) = Sum_{k=0..n} 3^k * binomial(n+k,k).
%F A385514 a(n) ~ 2^(2*n+1) * 3^(n+1) / (5*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 06 2025
%t A385514 Table[Sum[2^(n-k)*Binomial[2*n+1,k]*Binomial[2*n-k,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 05 2025 *)
%o A385514 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n+1, k)*binomial(2*n-k, n-k));
%o A385514 (Magma) [&+[2^(n-k) * Binomial(2*n+1,k) * Binomial(2*n-k,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 05 2025
%Y A385514 Cf. A385667, A385668.
%Y A385514 Cf. A383888.
%K A385514 nonn
%O A385514 0,2
%A A385514 _Seiichi Manyama_, Aug 04 2025