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A079520 Triangular array related to tennis ball problem, read by rows. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
Rows have been reversed.
LINKS
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
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

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Last modified April 21 13:26 EDT 2024. Contains 371870 sequences. (Running on oeis4.)