A387627 - OEIS

%I #16 Sep 04 2025 05:37:01

%S 1,3,7,27,83,263,855,2723,8731,27999,89663,287355,920771,2950263,

%T 9453607,30291667,97062123,311012623,996563855,3193247403,10231988371,

%U 32785923879,105054547063,336621829635,1078623042491,3456186066623,11074510391007,35485583833307

%N a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(2*n-2*k+1,2*k+1).

%H Vincenzo Librandi, <a href="/A387627/b387627.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,4,-4).

%F G.f.: (1+x-2*x^2)/((1+x-2*x^2)^2 - 4*x).

%F a(n) = 2*a(n-1) + 3*a(n-2) + 4*a(n-3) - 4*a(n-4).

%t Table[Sum[2^k*Binomial[2*n-2*k+1,2*k+1],{k,0,Floor[n/2]}],{n,0,40}] (* _Vincenzo Librandi_, Sep 04 2025 *)

%o (PARI) a(n) = sum(k=0, n\2, 2^k*binomial(2*n-2*k+1, 2*k+1));

%o (Magma) [&+[2^k* Binomial(2*n-2*k+1, 2*k+1): k in [0..Floor (n/2)]]: n in [0..35]]; // _Vincenzo Librandi_, Sep 04 2025

%Y Cf. A001653, A387628, A387629.

%Y Cf. A099511.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Sep 03 2025