A391676 - OEIS

%I #12 Dec 19 2025 17:19:59

%S 1,6,30,176,984,5232,27312,140160,707952,3531168,17428896,85249536,

%T 413738240,1994390016,9556560384,45550854144,216094481664,

%U 1020830461440,4804034860544,22529730809856,105326409295872,490983551209472,2282693571072000,10586924652920832

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

%H Vincenzo Librandi, <a href="/A391676/b391676.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: ((1-2*x-2*x^2)^2 + 8*x^3)/((1-2*x-2*x^2)^2 - 16*x^3)^(5/2).

%t Table[Sum[2^k* Binomial[k+2,2]*Binomial[k,n-k]^2,{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Dec 19 2025 *)

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

%o (Magma) [&+[2^k*Binomial(k+2,2)*Binomial(k, n-k)^2: k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Dec 19 2025

%Y Cf. A375276, A390612.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 16 2025