cp's OEIS Frontend

A386919 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(2*n-k,n-k).

%I A386919 #18 Aug 11 2025 10:20:22
%S A386919 1,6,58,624,7050,81926,969640,11624976,140708682,1715727090,
%T A386919 21043480458,259331888712,3208566672792,39830312782344,
%U A386919 495853462219600,6188170518911264,77393543796042570,969771226630919754,12172039459124750062,153006230384961477600,1925930502301667496250
%N A386919 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(2*n-k,n-k).
%H A386919 Vincenzo Librandi, <a href="/A386919/b386919.txt">Table of n, a(n) for n = 0..350</a>
%F A386919 a(n) = [x^n] (1+x)^(4*n)/(1-x)^(n+1).
%F A386919 a(n) = [x^n] 1/((1-x)^(2*n) * (1-2*x)^(n+1)).
%F A386919 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(3*n-k-1,n-k).
%F A386919 a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k-1,n-k).
%t A386919 Table[Sum[Binomial[4*n,k]*Binomial[2*n-k,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 10 2025 *)
%o A386919 (PARI) a(n) = sum(k=0, n, binomial(4*n, k)*binomial(2*n-k, n-k));
%o A386919 (Magma) [&+[Binomial(4*n,k) * Binomial(2*n-k,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 10 2025
%Y A386919 Cf. A066381, A386918, A386920.
%Y A386919 Cf. A385639.
%K A386919 nonn
%O A386919 0,2
%A A386919 _Seiichi Manyama_, Aug 08 2025