A079521 - OEIS

%I #7 Jan 17 2019 17:21:26

%S 0,1,2,3,5,4,10,16,13,6,31,47,45,25,8,105,158,145,96,41,10,343,501,

%T 500,340,175,61,12,1198,1752,1673,1226,676,288,85,14,4056,5808,5898,

%U 4326,2569,1205,441,113,16,14506,20868,20312,15608,9526,4836,1987,640,145,18,50350,71218,73000,55696,35448,18800,8418,3090,891,181,20

%N Triangular array related to tennis ball problem, read by rows.

%H G. C. Greubel, <a href="/A079521/b079521.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.4).

%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

%F g(t, r) = d(t)*(t*c(t))^r*(t*c(t)^3 + 2*r*c(t)) then the rows are calculated by the expansion of g(t,k) for n>=0, 0 <= k <= n. - _G. C. Greubel_, Jan 17 2019

%e 0.

%e 1,   2.

%e 3,   5,   4.

%e 10,  16,  13,  6.

%e 31,  47,  45,  25, 8.

%e 105, 158, 145, 96, 41, 10. ...

%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*c[t])^r*(t*c[t]^3 +2*r*c[t]); Table[SeriesCoefficient[Series[g[t, k], {t, 0, n}], n], {n, 0, 10}, {k, 0, n}] (* _G. C. Greubel_, Jan 17 2019 *)

%Y Leading diagonal gives A079522.

%Y Cf. A079513, A079520, A079521.

%K nonn,tabl

%O 0,3

%A _N. J. A. Sloane_, Jan 22 2003

%E Terms a(28) onward added by _G. C. Greubel_, Jan 17 2019