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A298567 a(n) = Sum_{k=0..2*n/3} C(n-k,2*k-n)^2.

%I A298567 #18 Jan 30 2020 21:29:18
%S A298567 1,0,1,1,1,4,2,9,10,17,37,41,102,136,251,450,667,1325,2011,3658,6246,
%T A298567 10293,18686,30461,54183,92169,157438,276414,466579,818256,1400509,
%U A298567 2419379,4202829,7208342,12556360,21621891,37480728,64965461,112227269
%N A298567 a(n) = Sum_{k=0..2*n/3} C(n-k,2*k-n)^2.
%F A298567 G.f.: 1/sqrt((1-x^2)^2+x^6-2*x^5-2*x^3).
%F A298567 D-finite with recurrence: n*a(n) -2*(n-1)*a(n-2)-(2*n-3)*a(n-3)+(n-2)*a(n-4) -(2*n-5)*a(n-5) +(n-3)*a(n-6) = 0. - _R. J. Mathar_, Jan 21 2020
%p A298567 A298567 := proc(n)
%p A298567     option remember;
%p A298567     if n < 7 then
%p A298567         op(n+1,[1, 0, 1, 1, 1, 4,2]) ;
%p A298567     else
%p A298567         -2*(n-1)*procname(n-2)-(2*n-3)*procname(n-3)+(n-2)*procname(n-4)
%p A298567             -(2*n-5)*procname(n-5)+(n-3)*procname(n-6) ;
%p A298567         -%/n ;
%p A298567     end if;
%p A298567 end proc: # _R. J. Mathar_, Jan 21 2020
%o A298567 (Maxima)
%o A298567 a(n):=sum(binomial(n-k,2*k-n)^2,k,0,2*n/3);
%Y A298567 Cf. A182878.
%K A298567 nonn
%O A298567 0,6
%A A298567 _Vladimir Kruchinin_, Jan 21 2018