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%I #8 Jan 17 2019 17:21:06
%S 12,284,5436,96768,1664184,28069444,467722524,7730252080,127023181352,
%T 2078332922360,33894711502744,551368536346176,8950922822411504,
%U 145068948446193428,2347940754318431196,37957946888159573968,613052225104703442120,9893099103451554441736
%N Related to tennis ball problem.
%H G. C. Greubel, <a href="/A079519/b079519.txt">Table of n, a(n) for n = 1..500</a>
%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344. (Table A.4)
%F Let f, g, S1 and S3 be given by f(t) = sqrt(1-4*t), g(t) = sqrt(1+4*t), S1(t) = (1+f(t)-2*f(t)^2)*(1- f(t))^5/(t^3*(f(t)^2-f(t))^2*(2+f(t)+g(t))^2), S3(t) = 4*(1-f(t))^2*(1 -g(t))^2*(f(t)^2-(1+2*t)*f(t)-(1-6*t)*g(t)+f(t)*g(t))/(t^3*(2+f(t)+ g(t))^2*(g(t)^2-f(t)-g(t)+ f(t)*g(t))^2). Now let W(t) be given by W(t) = (S1(t) + S1(-t) + S3(t) + S3(-t))/4. The g.f. is the expansion of W(t). - _G. C. Greubel_, Jan 17 2019
%e G.f. = 12*t^2 + 284*t^4 + 5436*t^6 + 96768*t^8 + ... - _G. C. Greubel_, Jan 17 2019
%t f[t_]:= Sqrt[1-4*t]; g[t_]:= Sqrt[1+4*t]; S1[t_]:= (1+f[t]-2*f[t]^2)*(1- f[t])^5/(t^3*(f[t]^2-f[t])^2*(2+f[t]+g[t])^2); S3[t_]:= 4*(1-f[t])^2*(1 -g[t])^2*(f[t]^2-(1+2*t)*f[t]-(1-6*t)*g[t]+f[t]*g[t])/(t^3*(2+f[t]+ g[t])^2*(g[t]^2-f[t]-g[t]+f[t]*g[t])^2); W[t_]:= (S1[t]+S1[-t]+S3[t]+ S3[-t])/4; Drop[CoefficientList[Series[W[t], {t, 0, 50}], t][[1 ;; ;; 2]], 1] (* _G. C. Greubel_, Jan 17 2019 *)
%Y Cf. A079513, A079514, A079515, A079516, A079517, A079518, A079520.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jan 22 2003
%E Terms a(5) onward added by _G. C. Greubel_, Jan 17 2019
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