A371777 - OEIS

%I #15 Apr 08 2024 09:08:34

%S 1,4,15,57,220,858,3368,13276,52479,207861,824527,3274395,13015081,

%T 51769813,206045841,820475513,3268499356,13025237058,51922543076,

%U 207034128448,825713206746,3293865399518,13142007903586,52443095356218,209304385553096,835459642193284

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

%F a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^n).

%F a(n) = binomial(2*(n+1), n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1+n/3, (4+n)/3, (5+n)/3], -1). - _Stefano Spezia_, Apr 06 2024

%F From _Vaclav Kotesovec_, Apr 08 2024: (Start)

%F Recurrence: n*a(n) = 3*(3*n-2)*a(n-1) - 6*(4*n-5)*a(n-2) + 8*(2*n-3)*a(n-3).

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

%F a(n) ~ 2^(2*n+1)/3.  (End)

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

%Y Cf. A371778, A371779, A371780.

%Y Cf. A144904, A371758, A371773.

%Y Cf. A032443.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 05 2024