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

0,6

Comments

Rows have been reversed.

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)

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

Example

0.
0, 1.
0, 1, 3.
0, 1, 4, 10.
0, 1, 5, 15, 31.
0, 1, 6, 21, 52, 105. ...

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 *)

Crossrefs

Leading diagonal gives A079522.

Cf. A079513.

Sequence in context: A139601 A213191 A352449 * A229001 A208981 A357892

Adjacent sequences: A079517 A079518 A079519 * A079521 A079522 A079523

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 22 2003

Extensions

Terms a(29) onward added by G. C. Greubel, Jan 17 2019

Status

approved