A143228 - OEIS

%I #15 Aug 28 2024 02:56:05

%S 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,

%T 121,15,15,30,45,75,105,165,225,22,22,44,66,110,154,242,330,484,30,30,

%U 60,90,150,210,330,450,660,900,42,42,84,126,210,294,462,630,924,1260,1764

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

%H G. C. Greubel, <a href="/A143228/b143228.txt">Rows n = 0..50 of the triangle, flattened</a>

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

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

%F Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A000041(n)*A087787(n). - _G. C. Greubel_, Aug 27 2024

%e First few rows of the triangle:

%e    1;

%e    1,  1;

%e    2,  2,  4;

%e    3,  3,  6,  9;

%e    5,  5, 10, 15, 25;

%e    7,  7, 14, 21, 35,  49;

%e   11, 11, 22, 33, 55,  77, 121;

%e   15, 15, 30, 45, 75, 105, 165, 225;

%e   ...

%e T(7,4) = 75 = p(7) * p(4) = 15 * 5.

%t Table[PartitionsP[n]*PartitionsP[k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 27 2024 *)

%o (Magma)

%o A143228:= func< n,k | NumberOfPartitions(n)*NumberOfPartitions(k) >;

%o [A143228(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 27 2024

%o (SageMath)

%o def A143215(n,k): return number_of_partitions(n)*number_of_partitions(k)

%o flatten([[A143215(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 27 2024

%Y Cf. A000041, A143229 (row sums).

%Y Main diagonal gives: A001255.

%K nonn,tabl

%O 0,4

%A _Gary W. Adamson_, Jul 31 2008