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A079521
Triangular array related to tennis ball problem, read by rows.
3
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
0,3
LINKS
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Fig. A.4).
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
EXAMPLE
0.
1, 2.
3, 5, 4.
10, 16, 13, 6.
31, 47, 45, 25, 8.
105, 158, 145, 96, 41, 10. ...
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 *)
CROSSREFS
Leading diagonal gives A079522.
Sequence in context: A347348 A254669 A227913 * A325549 A370895 A112060
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jan 22 2003
EXTENSIONS
Terms a(28) onward added by G. C. Greubel, Jan 17 2019
STATUS
approved