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A386939 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(3*n-k-1,n-k).

%I A386939 #18 Sep 03 2025 09:10:36
%S A386939 1,7,82,1083,15086,216566,3169636,47020371,704497750,10636206306,
%T A386939 161553957500,2465911305182,37791965926092,581171323026508,
%U A386939 8963417696439752,138590900605115779,2147571141595692390,33342454213792397930,518548824827926272268,8076888443386745743530
%N A386939 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(3*n-k-1,n-k).
%H A386939 Vincenzo Librandi, <a href="/A386939/b386939.txt">Table of n, a(n) for n = 0..1000</a>
%F A386939 a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(2*n).
%F A386939 a(n) = [x^n] 1/((1-x)^(n+2) * (1-2*x)^(2*n)).
%F A386939 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(2*n-k+1,n-k).
%F A386939 a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k-1,k) * binomial(2*n-k+1,n-k).
%t A386939 Table[Sum[Binomial[4*n+1, k]*Binomial[3*n-k-1,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A386939 (PARI) a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(3*n-k-1, n-k));
%o A386939 (Magma) [&+[Binomial(4*n+1,k) * Binomial(3*n-k-1,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A386939 Cf. A386834, A386938.
%K A386939 nonn,changed
%O A386939 0,2
%A A386939 _Seiichi Manyama_, Aug 10 2025