A376489 - OEIS

%I #9 Sep 26 2024 04:32:29

%S 1,0,1,1,4,7,22,49,143,359,1025,2742,7812,21666,62044,175927,507484,

%T 1460297,4243802,12340559,36108354,105839241,311551092,919000678,

%U 2719362502,8063263402,23967845874,71379427920,213010634136,636757780808,1906765570820

%N G.f. satisfies A(x) = 1 / (1 - x^2*A(x)^2 / (1 - x)).

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

%F D-finite with recurrence 8*n*(n+1)*a(n) -28*n*(n-1)*a(n-1) +2*(-9*n^2-n+14)*a(n-2) +(115*n^2-463*n+426)*a(n-3) +4*(-26*n^2+168*n-265)*a(n-4) +3*(3*n-13)*(3*n-14)*a(n-5)=0. - _R. J. Mathar_, Sep 26 2024

%p A376489 := proc(n)

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

%p end proc:

%p seq(A376489(n),n=0..70) ; # _R. J. Mathar_, Sep 26 2024

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

%Y Cf. A002212, A376490, A376491.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Sep 25 2024