A387928 - OEIS

%I #28 Oct 10 2025 11:03:22

%S 1,7,79,1000,13327,182887,2557684,36245608,518690095,7478672005,

%T 108474941239,1581019989028,23135973585796,339707845836340,

%U 5002372764627352,73846245417530320,1092507943002790255,16193897170945413505,240444441141909184045,3575491906219771528672

%N a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(2*n,k).

%H Vincenzo Librandi, <a href="/A387928/b387928.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = [x^n] (1-2*x)^n/(1-3*x)^(2*n+1).

%F a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(2*n+k,k).

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

%F a(n) = hypergeom([-2*n, -n], [1], 3). - _Stefano Spezia_, Sep 13 2025

%F a(n) = [x^n] ((1+x)^2 * (3+x))^n. - _Seiichi Manyama_, Sep 21 2025

%F a(n) ~ sqrt(3 + 19/sqrt(33)) * (59 + 11*sqrt(33))^n / (sqrt(Pi*n) * 2^(3*n + 3/2)). - _Vaclav Kotesovec_, Sep 21 2025

%F D-finite with recurrence 10*n*(2*n-1)*a(n) +(-327*n^2+407*n-150)*a(n-1) +8*(39*n^2-206*n+227)*a(n-2) +128*(n-2)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Sep 26 2025

%F 2*n*(2*n-1)*(11*n-14)*a(n) = (649*n^3-1475*n^2+964*n-180)*a(n-1) + 16*(n-1)*(2*n-3)*(11*n-3)*a(n-2) for n > 1. - _Seiichi Manyama_, Oct 10 2025

%t Table[Sum[ 3^k*Binomial[ n,k]*Binomial[2*n,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 20 2025 *)

%o (PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(2*n, k));

%o (Magma) [&+[3^k*Binomial(n, k)*Binomial(2*n,k): k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Sep 20 2025

%Y Cf. A069835, A387930, A387932.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 13 2025