A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

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

Offset

1,2

Comments

The row-reversed version is A168016.

Also see A168020.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, 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

T(n,k) = A000041(n/k) if k|n, else T(n,k)=0.

Sum_{k=1..n} T(n, k) = A047968(n).

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

T(2*n, n) = 2*A000012(n).

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

Example

Triangle begins:
==============================================
...... k: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12
==============================================
n=1 ..... 1,
n=2 ..... 2, 1,
n=3 ..... 3, 0, 1,
n=4 ..... 5, 2, 0, 1,
n=5 ..... 7, 0, 0, 0, 1,
n=6 .... 11, 3, 2, 0, 0, 1,
n=7 .... 15, 0, 0, 0, 0, 0, 1,
n=8 .... 22, 5, 0, 2, 0, 0, 0, 1,
n=9 .... 30, 0, 3, 0, 0, 0, 0, 0, 1,
n=10 ... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1,
n=11 ... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
n=12 ... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
...

Mathematica

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

Prog

(SageMath)
def A168021(n, k): return number_of_partitions(n/k) if (n%k)==0 else 0
flatten([[A168021(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Jan 12 2023

Crossrefs

Cf. A000007, A000012, A000041, A035377, A035444, A047968 (row sums).

Cf. A135010, A138121, A168016, A168017, A168018, A168019, A168020.

Sequence in context: A096798 A158902 A137587 * A137639 A239631 A113288

Adjacent sequences: A168018 A168019 A168020 * A168022 A168023 A168024

Keyword

easy,nonn,tabl

Author

Omar E. Pol, Nov 20 2009, Nov 21 2009

Extensions

Edited by Charles R Greathouse IV, Mar 23 2010

Status

approved