A393276 - OEIS

%I #18 Mar 08 2026 20:17:41

%S 1,0,0,0,2,6,12,20,36,78,182,408,866,1802,3804,8208,17874,38802,83778,

%T 180660,390636,847490,1841868,4004124,8704740,18932398,41213490,

%U 89798064,195789922,427093530,932052968,2034966160,4445125152,9714318306,21238410158,46451025612,101630144708

%N a(n) = Sum_{k=0..floor(n/4)} binomial(2*k,k) * binomial(n-k-1,n-4*k).

%H Vincenzo Librandi, <a href="/A393276/b393276.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/sqrt(1 - 4*x^4/(1-x)^3).

%F D-finite with recurrence: (2 + 4*n)*a(n) + (-11 - 3*n)*a(n + 1) + (-8 - 4*n)*a(n + 2) + (18 + 6*n)*a(n + 3) + (-4*n - 16)*a(n + 4) + (n + 5)*a(n + 5) = 0. - _Robert Israel_, Mar 08 2026

%p f:= gfun:-rectoproc({(2 + 4*n)*a(n) + (-11 - 3*n)*a(n + 1) + (-8 - 4*n)*a(n + 2) + (18 + 6*n)*a(n + 3) + (-4*n - 16)*a(n + 4) + (n + 5)*a(n + 5), a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 2},a(n), remember):

%p map(f, [$0..35]); # _Robert Israel_, Mar 08 2026

%t Table[Sum[Binomial[2*k,k]*Binomial[n-k-1,n-4*k],{k,0,Floor[n/4]}],{n,0,35}] (* _Vincenzo Librandi_, Feb 11 2026 *)

%o (PARI) a(n) = sum(k=0, n\4, binomial(2*k, k)*binomial(n-k-1, n-4*k));

%o (Magma) [&+[Binomial(2*k, k)* Binomial(n-k-1, n-4*k) : k in [0..Floor(n/4)]] : n in [0..43] ]; // _Vincenzo Librandi_, Feb 11 2026

%Y Partial sums are A393246.

%Y Cf. A360310, A393275, A393277.

%Y Cf. A026585, A376809.

%K nonn,easy

%O 0,5

%A _Seiichi Manyama_, Feb 08 2026