A079521 Triangular array related to tennis ball problem, read by rows.

0, 1, 2, 3, 5, 4, 10, 16, 13, 6, 31, 47, 45, 25, 8, 105, 158, 145, 96, 41, 10, 343, 501, 500, 340, 175, 61, 12, 1198, 1752, 1673, 1226, 676, 288, 85, 14, 4056, 5808, 5898, 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

Offset: 0

Author

N. J. A. Sloane, Jan 22 2003

Keywords

• nonn
• tabl

Examples

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

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.4).

Crossrefs

Leading diagonal gives A079522.

Cf. A079513, A079520, A079521.

Programs

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*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 *)

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*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

Extensions

Terms a(28) onward added by G. C. Greubel, Jan 17 2019