A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.

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

Offset

1,3

Links

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

Formula

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

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

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

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

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

Sum_{k=2..n-1} T(n, k) = A168111(n-1). (End)

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A000007, A000041, A035363, A035444, A047968, A135010.

Cf. A138121, A168014, A168015, A168020, A168021, A168111.

Sequence in context: A247492 A113214 A280494 * A342301 A029323 A194849

Adjacent sequences: A168013 A168014 A168015 * A168017 A168018 A168019

Keyword

easy,nonn,tabl

Author

Omar E. Pol, Nov 21 2009

Extensions

Edited and extended by Max Alekseyev, May 07 2010

Status

approved