A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.

1, 2, 0, 3, 1, 0, 5, 0, 0, 0, 7, 2, 1, 0, 0, 11, 0, 0, 0, 0, 0, 15, 3, 0, 1, 0, 0, 0, 22, 0, 2, 0, 0, 0, 0, 0, 30, 5, 0, 0, 1, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 11, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Omar E. Pol, Nov 20 2009

Keywords

• easy
• nonn
• tabl

Comments

In the square array, note that the column k starts with k-1 zeros. Then list each partition number of positive integers followed by k-1 zeros. See A000041, which is the main entry for this sequence.

Examples

			The array, A(n, k), begins:
   n | k = 1   2   3   4   5   6   7   8   9  10  11  12
  ---+--------------------------------------------------
   1 |     1   0   0   0   0   0   0   0   0   0   0   0
   2 |     2   1   0   0   0   0   0   0   0   0   0   0
   3 |     3   0   1   0   0   0   0   0   0   0   0   0
   4 |     5   2   0   1   0   0   0   0   0   0   0   0
   5 |     7   0   0   0   1   0   0   0   0   0   0   0
   6 |    11   3   2   0   0   1   0   0   0   0   0   0
   7 |    15   0   0   0   0   0   1   0   0   0   0   0
   8 |    22   5   0   2   0   0   0   1   0   0   0   0
   9 |    30   0   3   0   0   0   0   0   1   0   0   0
  10 |    42   7   0   0   2   0   0   0   0   1   0   0
  11 |    56   0   0   0   0   0   0   0   0   0   1   0
  12 |    77  11   5   3   0   2   0   0   0   0   0   1
  ...
Antidiagonal triangle, T(n,k), begins as:
   1;
   2, 0;
   3, 1, 0;
   5, 0, 0, 0;
   7, 2, 1, 0, 0;
  11, 0, 0, 0, 0, 0;
  15, 3, 0, 1, 0, 0, 0;
  22, 0, 2, 0, 0, 0, 0, 0;
  30, 5, 0, 0, 1, 0, 0, 0, 0;
  42, 0, 0, 0, 0, 0, 0, 0, 0, 0;
		

Links

• G. C. Greubel, Antidiagonals n = 1..50, flattened
• Omar E. Pol, Illustration of the shell model of partitions (2D and 3D)
• Omar E. Pol, Illustration of the shell model of partitions (2D view)
• Omar E. Pol, Illustration of the shell model of partitions (3D view)

Crossrefs

Cf. A000007, A000041, A035377, A035444, A135010, A138121.

Cf. A168016, A168017, A168018, A168019, A168021. [From Omar E. Pol, Nov 23 2009]

Programs

Mathematica

T[n_, k_]:= If[IntegerQ[(n-k+1)/k], PartitionsP[(n-k+1)/k], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

SageMath

def A168020(n,k): return number_of_partitions((n-k+1)/k) if ((n-k+1)%k)==0 else 0
flatten([[A168020(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

Formula

A(n, k) = A000041(n/k) if k divides n, otherwise A(n, k) = 0 (array).

A(n, 1) = A(n*k, k) = A000041(n).

From G. C. Greubel, Jan 12 2023: (Start)

T(n, k) = A000041((n-k+1)/k) if k divides (n-k+1), otherwise T(n, k) = 0 (triangle).

T(n, 1) = A000041(n).

T(2*n, n) = 2*A000007(n-1), n >= 1. (End)

Extensions

Edited by Omar E. Pol, Nov 21 2009

Edited by Charles R Greathouse IV, Mar 23 2010

Edited by Max Alekseyev, May 07 2010