A079520 Triangular array related to tennis ball problem, read by rows.
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, 0, 1, 8, 36, 116, 301, 644, 1198, 0, 1, 9, 45, 161, 462, 1106, 2304, 4056, 0, 1, 10, 55, 216, 678, 1784, 4088, 8144, 14506, 0, 1, 11, 66, 282, 960, 2744, 6832, 14976, 29482, 50350
Offset: 0
Examples
0. 0, 1. 0, 1, 3. 0, 1, 4, 10. 0, 1, 5, 15, 31. 0, 1, 6, 21, 52, 105. ...
Links
- G. C. Greubel, Rows n=0..100 of triangle, flattened
- D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Fig. A.3)
Programs
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Mathematica
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 *)
Formula
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
Extensions
Terms a(29) onward added by G. C. Greubel, Jan 17 2019
Comments