A393277 - OEIS

%I #19 Mar 08 2026 20:14:56

%S 1,0,0,0,2,4,6,8,16,36,74,136,248,476,946,1864,3606,6948,13506,26456,

%T 51856,101396,198158,387976,761170,1494636,2935278,5766264,11335628,

%U 22301596,43901474,86457032,170330356,335718772,661994394,1305896760,2577024428,5087143484,10045529338

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

%H Vincenzo Librandi, <a href="/A393277/b393277.txt">Table of n, a(n) for n = 0..1000</a>

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

%F D-finite with recurrence: (4 + 4*n)*a(n) + (-4*n - 12)*a(n + 1) + (-2 - n)*a(n + 2) + (9 + 3*n)*a(n + 3) + (-3*n - 12)*a(n + 4) + (n + 5)*a(n + 5) = 0. - _Robert Israel_, Mar 08 2026

%p f:= gfun:-rectoproc({(4 + 4*n)*a(n) + (-4*n - 12)*a(n + 1) + (-2 - n)*a(n + 2) + (9 + 3*n)*a(n + 3) + (-3*n - 12)*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..40]); # _Robert Israel_, Mar 08 2026

%t Table[Sum[Binomial[2*k,k]*Binomial[n-2*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-2*k-1, n-4*k));

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

%Y Partial sums are A098482.

%Y Cf. A360310, A393275, A393276.

%K nonn,easy

%O 0,5

%A _Seiichi Manyama_, Feb 08 2026