A168120 -id:A168120 - cp's OEIS frontend

A168019 Square array A(n,k) read by antidiagonals, in which row n lists the number of partitions of n into parts divisible by k+1.

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

Offset: 0

Omar E. Pol, Nov 21 2009

Note that column k lists each partition number A000041 followed by k zeros. See also A168020 and A168021.

Let A(n,k) denote the number of partitions of n into parts divisible by k+1. Let p(n) denote the number of partitions of n. If k+1 is a divisor of n then A(n,k) = p(n/(k+1)) otherwise A(n,k) = 0. [Conjectured by Omar E. Pol, Nov 25 2009] - this is trivial, just divide each part size by k - Franklin T. Adams-Watters, May 14 2010.

			The array, A(n, k), begins:
==================================================
... Column k: 0.. 1. 2. 3. 4. 5. 6. 7. 8. 9 10 11
. Row ...........................................
...n ............................................
==================================================
.. 0 ........ 1,  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
.. 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;
   1, 1;
   2, 0, 1;
   3, 1, 0, 1;
   5, 0, 0, 0, 1;
   7, 2, 1, 0, 0, 1;
  11, 0, 0, 0, 0, 0, 1;
  15, 3, 0, 1, 0, 0, 0, 1;
  22, 0, 2, 0, 0, 0, 0, 0, 1;
  30, 5, 0, 0, 1, 0, 0, 0, 0, 1;
  42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A000041, A035363, A083710, A135010.

Cf. A168016, A168120, A168021, A168121.

T[n_, k_]:= If[IntegerQ[(n-k)/(k+1)], PartitionsP[(n-k)/(k+1)], 0];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 13 2023 *)

def A168019(n,k): return number_of_partitions((n-k)/(k+1)) if ((n-k)%(k+1))==0 else 0
flatten([[A168019(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 13 2023

From G. C. Greubel, Jan 13 2023: (Start)
A(n, k) = A000041(n/(k+1)) if (k+1)|n, otherwise 0 (array).
T(n, k) = A000041((n-k)/(k+1)) if (k+1)|(n-k), otherwise 0 (antidiagonals).
A(n, 0) = T(n, 0) = A000041(n).
T(2*n, n) = A(n, n) = A000007(n).
Sum_{k=0..n} T(n, k) = A083710(n+1). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

Edited by Franklin T. Adams-Watters, May 14 2010

A168121 Triangle T(n,k) read by rows in which column k lists each number A000009 followed by k-1 zeros, for k>0.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 3, 0, 0, 0, 1, 4, 2, 1, 0, 0, 1, 5, 0, 0, 0, 0, 0, 1, 6, 2, 0, 1, 0, 0, 0, 1, 8, 0, 2, 0, 0, 0, 0, 0, 1, 10, 3, 0, 0, 1, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 15, 4, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Dec 02 2009

			Triangle begins:
==================================================
... Column k: 1. 2. 3. 4. 5. 6. 7. 8. 9 10 11 12
. Row ...........................................
...n ............................................
==================================================
.. 1 ........ 1,
.. 2 ........ 1, 1,
.. 3 ........ 2, 0, 1,
.. 4 ........ 2, 1, 0, 1,
.. 5 ........ 3, 0, 0, 0, 1,
.. 6 ........ 4, 2, 1, 0, 0, 1,
.. 7 ........ 5, 0, 0, 0, 0, 0, 1,
.. 8 ........ 6, 2, 0, 1, 0, 0, 0, 1,
.. 9 ........ 8, 0, 2, 0, 0, 0, 0, 0, 1,
. 10 ....... 10, 3, 0, 0, 1, 0, 0, 0, 0, 1,
. 11 ....... 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
. 12 ....... 15, 4, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1,
...

Cf. A000009, A168021, A168120.

Edited by Charles R Greathouse IV, Mar 23 2010

cp's OEIS Frontend

A168019 Square array A(n,k) read by antidiagonals, in which row n lists the number of partitions of n into parts divisible by k+1.

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