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

1,2

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.

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)

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)

Example

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;

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A202064 A144955 A225624 * A321878 A225084 A238345

Adjacent sequences: A168017 A168018 A168019 * A168021 A168022 A168023

Keyword

easy,nonn,tabl

Author

Omar E. Pol, Nov 20 2009

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

Status

approved