A079513 Triangular array (a Riordan array) related to tennis ball problem, read by rows.

1, 0, 1, 1, 1, 1, 0, 3, 2, 1, 6, 6, 6, 3, 1, 0, 22, 16, 10, 4, 1, 53, 53, 53, 31, 15, 5, 1, 0, 211, 158, 105, 52, 21, 6, 1, 554, 554, 554, 343, 185, 80, 28, 7, 1, 0, 2306, 1752, 1198, 644, 301, 116, 36, 8, 1, 6362, 6362, 6362, 4056, 2304, 1106, 462, 161, 45, 9, 1

Offset: 0

Author

N. J. A. Sloane, Jan 22 2003

Keywords

• nonn
• tabl

Comments

Riordan array (2/(2-x*c(x)+x*c(-x)), x*c(x)), with c(x) the g.f. of Catalan numbers (A000108). - Ralf Stephan, Dec 29 2013

Examples

			Triangle starts
     1;
     0,    1;
     1,    1,    1;
     0,    3,    2,    1;
     6,    6,    6,    3,    1;
     0,   22,   16,   10,    4,    1;
    53,   53,   53,   31,   15,    5,   1;
     0,  211,  158,  105,   52,   21,   6,   1;
   554,  554,  554,  343,  185,   80,  28,   7,  1;
     0, 2306, 1752, 1198,  644,  301, 116,  36,  8, 1;
  6362, 6362, 6362, 4056, 2304, 1106, 462, 161, 45, 9, 1;
		

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 (Table A.2).

Crossrefs

First column is A066357 interspersed with 0's, 2nd column gives A079514.

Cf. A079514, A079515, A079516, A079517, A079518, A079519, 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; Table[SeriesCoefficient[Series[g[t, k], {t, 0, n}], n], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 16 2019 *)

Extensions

Edited and more terms added by Ralf Stephan, Dec 29 2013