A079519 Related to tennis ball problem.
12, 284, 5436, 96768, 1664184, 28069444, 467722524, 7730252080, 127023181352, 2078332922360, 33894711502744, 551368536346176, 8950922822411504, 145068948446193428, 2347940754318431196, 37957946888159573968, 613052225104703442120, 9893099103451554441736
Offset: 1
Keywords
Examples
G.f. = 12*t^2 + 284*t^4 + 5436*t^6 + 96768*t^8 + ... - _G. C. Greubel_, Jan 17 2019
Links
- G. C. Greubel, Table of n, a(n) for n = 1..500
- D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Table A.4)
Programs
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Mathematica
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 *)
Formula
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
Extensions
Terms a(5) onward added by G. C. Greubel, Jan 17 2019