A143228 Triangle read by rows, T(n,k) = p(n) * p(k), where p(n) = the number of partitions of n, for 0 <= k <= n.

1, 1, 1, 2, 2, 4, 3, 3, 6, 9, 5, 5, 10, 15, 25, 7, 7, 14, 21, 35, 49, 11, 11, 22, 33, 55, 77, 121, 15, 15, 30, 45, 75, 105, 165, 225, 22, 22, 44, 66, 110, 154, 242, 330, 484, 30, 30, 60, 90, 150, 210, 330, 450, 660, 900, 42, 42, 84, 126, 210, 294, 462, 630, 924, 1260, 1764

Offset: 0

Author

Gary W. Adamson, Jul 31 2008

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  4;
   3,  3,  6,  9;
   5,  5, 10, 15, 25;
   7,  7, 14, 21, 35,  49;
  11, 11, 22, 33, 55,  77, 121;
  15, 15, 30, 45, 75, 105, 165, 225;
  ...
T(7,4) = 75 = p(7) * p(4) = 15 * 5.
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000041, A143229 (row sums).

Main diagonal gives: A001255.

Programs

Magma

A143228:= func< n,k | NumberOfPartitions(n)*NumberOfPartitions(k) >;
[A143228(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2024

Mathematica

Table[PartitionsP[n]*PartitionsP[k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2024 *)

SageMath

def A143215(n,k): return number_of_partitions(n)*number_of_partitions(k)
flatten([[A143215(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 27 2024

Formula

T(n, 0) = A000041(n) (left border).

Sum_{k=0..n} T(n, k) = A143229(n) (row sums).

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A000041(n)*A087787(n). - G. C. Greubel, Aug 27 2024