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

0,8

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

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).

Example

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;

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 *)

Crossrefs

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

Cf. A079514, A079515, A079516, A079517, A079518, A079519, A079520, A079521.

Sequence in context: A208520 A114155 A192018 * A060408 A267121 A208518

Adjacent sequences: A079510 A079511 A079512 * A079514 A079515 A079516

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 22 2003

Extensions

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

Status

approved