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
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