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%I #7 Jan 17 2019 17:21:20
%S 0,0,1,0,1,3,0,1,4,10,0,1,5,15,31,0,1,6,21,52,105,0,1,7,28,80,185,343,
%T 0,1,8,36,116,301,644,1198,0,1,9,45,161,462,1106,2304,4056,0,1,10,55,
%U 216,678,1784,4088,8144,14506,0,1,11,66,282,960,2744,6832,14976,29482,50350
%N Triangular array related to tennis ball problem, read by rows.
%C Rows have been reversed.
%H G. C. Greubel, <a href="/A079520/b079520.txt">Rows n=0..100 of triangle, flattened</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. (Fig. A.3)
%F Let c, d, and g be given by: c(t) = (1-sqrt(1-4*t))/(2*t), d(t) = (1-(1+2*t)*sqrt(1-4*t) -(1- 2*t)*sqrt(1+4*t) +sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r+1)*c(t)^(r+3). The rows of the triangle are calculated by the expansion of g(t, n-k) for n>=0, 0 <= k <= n. - _G. C. Greubel_, Jan 17 2019
%e 0.
%e 0, 1.
%e 0, 1, 3.
%e 0, 1, 4, 10.
%e 0, 1, 5, 15, 31.
%e 0, 1, 6, 21, 52, 105. ...
%t c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1- 2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1) *c[t]^(r+3); Table[SeriesCoefficient[Series[g[t, n-k], {t, 0, n}], n], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Jan 17 2019 *)
%Y Leading diagonal gives A079522.
%Y Cf. A079513.
%K nonn,tabl
%O 0,6
%A _N. J. A. Sloane_, Jan 22 2003
%E Terms a(29) onward added by _G. C. Greubel_, Jan 17 2019
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