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%I #19 Mar 06 2022 08:55:10
%S 1,0,2,12,50,280,1442,7812,42338,232176,1280642,7103932,39579410,
%T 221340808,1241708834,6984796852,39382895810,222512915680,
%U 1259482604546,7140546372204,40541480041970,230480474747640,1311841695315362,7474722997813732,42631911134818850
%N a(n) = Sum_{k=0..n} (-1)^(n+k)*A111516(n,k).
%F O.g.f.: ((-3*x^2 + 5*x + 4)*sqrt(x^2 - 6*x + 1) - 3*x^3 + 2*x^2 + 5*x)/(sqrt(x^2 - 6*x + 1)*(6*x^2 + 10*x + 4)).
%F a(n) ~ sqrt(9*sqrt(2) - 8) * (1 + sqrt(2))^(2*n) / (14*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 14 2021
%F D-finite with recurrence -2*(n-1)*(56*n^2-328*n+467)*a(n) +(392*n^3-3024*n^2+7501*n-5979)*a(n-1) +(1400*n^3-10328*n^2+24
%F 187*n-17824)*a(n-2) +(728*n^3-5216*n^2+11919*n-8571)*a(n-3) -3*(n-4)*(56*n^2-216*n+195)*a(n-4)=0. - _R. J. Mathar_, Mar 06 2022
%F D-finite with recurrence 10*(-n+1)*a(n) +(31*n-53)*a(n-1) +(145*n-274)*a(n-2) +2*(47*n-69)*a(n-3) +2*(-32*n+179)*a(n-4) +3*(-15*n+77)*a(n-5) +9*(n-6)*a(n-6)=0. - _R. J. Mathar_, Mar 06 2022
%e a(3) = (-1)^3*(1 - 7 + 12 - 18) = 12.
%e a(4) = (-1)^4*(1 - 15 + 32 - 56 + 88) = 50.
%o (PARI) G(n,k) = if (k==0, 1, sum(j=1,n, binomial(n,j)*binomial(k+j-2,j-1))); \\ A111516
%o a(n) = sum(k=0, n, (-1)^(n+k)*G(n,k)); \\ _Michel Marcus_, Feb 14 2021
%Y Cf. A111516.
%K nonn
%O 0,3
%A _Petros Hadjicostas_, Feb 14 2021
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