A390705 - OEIS

%I #13 Nov 18 2025 03:35:58

%S 1,1,6,36,217,1332,8295,52221,331518,2118558,13611144,87833781,

%T 568893156,3696245515,24080248485,157245263196,1028928699129,

%U 6744977287656,44287036450035,291206172153066,1917300157617312,12638390384014515,83398707478813197,550874308524107364

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

%H Vincenzo Librandi, <a href="/A390705/b390705.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = Sum_{k=0..n} (-3)^k * binomial(3*n+k+1,n-k).

%t Table[Sum[(-3)^k*Binomial[3*n +k+1,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Nov 18 2025 *)

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

%o (Magma) [&+[(-3)^k*Binomial(3*n+k+1, n-k): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Nov 18 2025

%Y Cf. A371871, A390706.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 15 2025