A259975 Irregular triangle read by rows: T(n,k) = number of ways of placing n balls into k boxes in such a way that any two adjacent boxes contain at least 4 balls.

1, 1, 1, 1, 1, 5, 1, 1, 6, 4, 1, 7, 9, 1, 8, 16, 1, 9, 25, 15, 1, 1, 10, 35, 40, 8, 1, 11, 46, 76, 31, 1, 12, 58, 124, 85, 1, 13, 71, 185, 190, 35, 1, 1, 14, 85, 260, 360, 154, 13, 1, 15, 100, 350, 610, 424, 76, 1, 16, 116, 456, 956, 930, 295

Offset: 0

Author

N. J. A. Sloane, Jul 12 2015

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  1;
  1;
  1;
  1,  5,   1;
  1,  6,   4;
  1,  7,   9;
  1,  8,  16;
  1,  9,  25,  15,   1;
  1, 10,  35,  40,   8;
  1, 11,  46,  76,  31;
  1, 12,  58, 124,  85;
  1, 13,  71, 185, 190,  35,  1;
  1, 14,  85, 260, 360, 154, 13;
  1, 15, 100, 350, 610, 424, 76;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• R. P. Boas & N. J. A. Sloane, Correspondence, 1974
• D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
• M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.

Crossrefs

Columns: A004120, A005337, A005338, A005339, A005340.

Row sums give A257666.

Programs

Maple

b:= proc(n, v) option remember; expand(`if`(n=0,
      `if`(v=0, 1+x, 1), add(x*b(n-j, max(0, 4-j)), j=v..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 12 2015

Mathematica

b[n_, v_] := b[n, v] = Expand[If[n == 0, If[v == 0, 1+x, 1], Sum[x*b[n-j, Max[0, 4-j]], {j, v, n}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Jul 12 2015