A360219 - OEIS

%I #42 Jul 13 2024 13:47:05

%S 1,2,6,20,68,240,864,3152,11616,43136,161152,604992,2280416,8624832,

%T 32714688,124399488,474066560,1810053120,6922776576,26517173760,

%U 101710338048,390603984896,1501732753408,5779500226560,22263437981184,85835254221824,331193445626880

%N a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).

%C Diagonal of rational function 1/(1 - x - y + x^4*y^3). - _Seiichi Manyama_, Mar 23 2023

%H Seiichi Manyama, <a href="/A360219/b360219.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%p A360219 := proc(n)

%p     add((-1)^k*binomial(n-3*k,k)*binomial(2*(n-3*k),n-3*k),k=0..n/3) ;

%p end proc:

%p seq(A360219(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023

%o (PARI) a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x^3)))

%Y Cf. A157004, A360267, A374599.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 31 2023