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A372018 G.f. A(x) satisfies A(x) = ( 1 + 4*x*A(x)/(1 - x*A(x)) )^(1/2).

%I A372018 #12 Apr 22 2024 12:45:55
%S A372018 1,2,4,10,30,98,336,1194,4360,16258,61644,236938,921102,3615330,
%T A372018 14307312,57024426,228701646,922283522,3737497980,15212318730,
%U A372018 62160993642,254909413218,1048717979424,4327273358250,17903826642780,74260741616514,308724721176676
%N A372018 G.f. A(x) satisfies A(x) = ( 1 + 4*x*A(x)/(1 - x*A(x)) )^(1/2).
%F A372018 a(n) = (1/(n+1)) * Sum_{k=0..n} 4^k * binomial(n/2+1/2,k) * binomial(n-1,n-k).
%F A372018 D-finite with recurrence n*(n+1)*(n-2)*a(n) -6*(n-2)*(3*n^2-6*n+1)*a(n-2) -27*n*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Apr 22 2024
%F A372018 Conjecture: a(2n+1) = 2*A371364(). - _R. J. Mathar_, Apr 22 2024
%p A372018 A372018 := proc(n)
%p A372018     add(4^k*binomial((n+1)/2,k)*binomial(n-1,k-1),k=0..n) ;
%p A372018     %/(n+1) ;
%p A372018 end proc:
%p A372018 seq(A372018(n),n=0..60) ; # _R. J. Mathar_, Apr 22 2024
%o A372018 (PARI) a(n) = sum(k=0, n, 4^k*binomial(n/2+1/2, k)*binomial(n-1, n-k))/(n+1);
%Y A372018 Cf. A372019, A372020.
%K A372018 nonn
%O A372018 0,2
%A A372018 _Seiichi Manyama_, Apr 15 2024