A089177 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n))) giving number of non-squashing partitions of n into k parts.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 4, 1, 1, 5, 6, 2, 1, 6, 9, 4, 1, 7, 12, 6, 1, 8, 16, 10, 1, 1, 9, 20, 14, 2, 1, 10, 25, 20, 4, 1, 11, 30, 26, 6, 1, 12, 36, 35, 10, 1, 13, 42, 44, 14, 1, 14, 49, 56, 20, 1, 15, 56, 68, 26, 1, 16, 64, 84, 36, 1, 1, 17, 72, 100, 46, 2, 1, 18, 81, 120, 60, 4, 1

Offset

0,5

Comments

T(n,k) = A181322(n,k) - A181322(n,k-1) for n>0. - Alois P. Heinz, Jan 25 2014

Links

Alois P. Heinz, Rows n = 0..1002, flattened

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Formula

Row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor(n/2) shifted one place right).

G.f. for column k (k >= 2): x^(2^(k-2))/((1-x)*Product_j=1..k-2} (1-x^(2^j))).

Example

Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  2;
  1, 4,  4,  1;
  1, 5,  6,  2;
  1, 6,  9,  4;
  1, 7, 12,  6;
  1, 8, 16, 10,  1;

Maple

T:= proc(n) option remember;
     `if`(n=0, 1, zip((x, y)-> x+y, [T(n-1)], [0, T(floor(n/2))], 0)[])
    end:
seq(T(n), n=0..25);  # Alois P. Heinz, Apr 01 2012

Mathematica

row[0] = {1}; row[1] = {1, 1}; row[n_] := row[n] = Plus @@ PadRight[ {row[n-1], Join[{0}, row[Floor[n/2]]]} ]; Table[row[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Jan 31 2014 *)

Crossrefs

Cf. A089178. Columns give A002620, A008804, A088932, A088954. Row sums give A000123.

Sequence in context: A226130 A137569 A266715 * A348445 A023996 A307998

Adjacent sequences: A089174 A089175 A089176 * A089178 A089179 A089180

Keyword

nonn,tabf,look,easy

Author

N. J. A. Sloane, Dec 08 2003

Extensions

More terms from Alford Arnold, May 22 2004

Status

approved