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A306463 a(n) = Sum_{k=0..n} Sum_{m=0..floor(k/2)} binomial(k-m, m)*binomial(n-k, k-m)^2.

%I A306463 #24 Jan 30 2020 21:29:18
%S A306463 1,1,2,6,15,37,98,262,699,1883,5110,13918,38045,104355,287028,791320,
%T A306463 2186209,6051113,16776022,46577806,129491865,360432855,1004332322,
%U A306463 2801307498,7820572153,21851390549,61101872126,170977916730,478755116117,1341389394715,3760507521800
%N A306463 a(n) = Sum_{k=0..n} Sum_{m=0..floor(k/2)} binomial(k-m, m)*binomial(n-k, k-m)^2.
%H A306463 Seiichi Manyama, <a href="/A306463/b306463.txt">Table of n, a(n) for n = 0..2202</a>
%F A306463 G.f.: 1/sqrt(x^6 + 2*x^5 - x^4 - 4*x^3 - x^2 - 2*x + 1).
%F A306463 D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +(-n+1)*a(n-2) +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 16 2020
%o A306463 (Maxima)
%o A306463 a(n):=sum(sum(binomial(k-m,m)*binomial(n-k,k-m)^2,m,0,k/2),k,0,n);
%o A306463 (PARI) a(n) = sum(k=0, n, sum(m=0, k\2, binomial(k-m, m)*binomial(n-k, k-m)^2)); \\ _Michel Marcus_, Feb 18 2019
%o A306463 (PARI) N=66; x='x+O('x^N); Vec(1/sqrt(x^6+2*x^5-x^4-4*x^3-x^2-2*x+1)) \\ _Seiichi Manyama_, Feb 20 2019
%Y A306463 Cf. A008459, A306504.
%K A306463 nonn
%O A306463 0,3
%A A306463 _Vladimir Kruchinin_, Feb 17 2019