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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.
2
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
0,4
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
EXAMPLE
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.
MATHEMATICA
Table[PartitionsP[n]*PartitionsP[k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 27 2024 *)
PROG
(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
(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
CROSSREFS
Cf. A000041, A143229 (row sums).
Main diagonal gives: A001255.
Sequence in context: A303691 A205678 A128590 * A143211 A361757 A209755
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jul 31 2008
STATUS
approved