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a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-3*k,n-2*k).
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%I #14 Mar 02 2023 09:38:37

%S 1,2,5,17,61,221,812,3021,11344,42899,163146,623320,2390653,9198879,

%T 35494701,137290466,532149805,2066501909,8038146035,31312535610,

%U 122140123201,477002869614,1864912495716,7298427590543,28588888586743,112080607196843,439744801379594

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

%F G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2 * c(x)) ), where c(x) is the g.f. of A000108.

%F a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 18 2023

%F D-finite with recurrence 2*n*a(n) +(-5*n+2)*a(n-1) +(-11*n+12)*a(n-2) +2*(-n+5)*a(n-3) +(-7*n+2)*a(n-4) +2*(-2*n+5)*a(n-5)=0. - _R. J. Mathar_, Mar 02 2023

%p A360211 := proc(n)

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

%p end proc:

%p seq(A360211(n),n=0..40) ; # _R. J. Mathar_, Mar 02 2023

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+2*x^2/(1+sqrt(1-4*x)))))

%Y Cf. A005317, A024718, A176332, A360185.

%Y Cf. A000108, A176287.

%Y Cf. A026641, A360212.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 30 2023