A360267 - OEIS

%I #23 Jul 13 2024 13:47:10

%S 1,2,6,20,72,264,984,3712,14136,54224,209200,810912,3155616,12320512,

%T 48239232,189336192,744722400,2934759360,11584470336,45796087680,

%U 181285742592,718498695424,2850802065152,11322567705600,45011437903104,179088911779328

%N a(n) = Sum_{k=0..floor(n/4)} 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="/A360267/b360267.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).

%F a(n) ~ 1 / (2*sqrt((1 - 3*r)*Pi*n) * r^n), where r = 0.2463187933841190115229... is the positive real root of the equation -1 + 4*r + 4*r^4 = 0. - _Vaclav Kotesovec_, Mar 23 2023

%t Table[Sum[Binomial[n-3k,k]Binomial[2(n-3k),n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* _Harvey P. Dale_, May 27 2024 *)

%o (PARI) a(n) = sum(k=0, n\4, 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. A006139, A360266.

%Y Cf. A360219, A374599.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 31 2023