A390407 - OEIS

%I #24 Nov 11 2025 02:41:05

%S 1,5,35,274,2261,19228,166726,1465399,13008995,116374621,1047384196,

%T 9473049468,86028257666,783933555215,7164492260569,65642607902166,

%U 602754037713365,5545405993706068,51105791195006851,471704973073493801,4359800639429095004

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

%H Vincenzo Librandi, <a href="/A390407/b390407.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/((1-4*x*g^3) * (1-x*g^2)) where g = 1+x*g^4 is the g.f. of A002293.

%F a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n+k+2,n-k).

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

%t Table[Sum[Binomial[4*n-2*k,n-k],{k,0,n}],{n,0,20}] (* _Vincenzo Librandi_, Nov 07 2025 *)

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

%Y Cf. A052203, A066381, A225612, A389361, A389239, A390408, A390409, A390410.

%Y Cf. A006134, A388043.

%Y Cf. A002293.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 04 2025