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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.

%I A143228 #15 Aug 28 2024 02:56:05
%S A143228 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 A143228 121,15,15,30,45,75,105,165,225,22,22,44,66,110,154,242,330,484,30,30,
%U A143228 60,90,150,210,330,450,660,900,42,42,84,126,210,294,462,630,924,1260,1764
%N 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.
%H A143228 G. C. Greubel, <a href="/A143228/b143228.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A143228 T(n, 0) = A000041(n) (left border).
%F A143228 Sum_{k=0..n} T(n, k) = A143229(n) (row sums).
%F A143228 Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A000041(n)*A087787(n). - _G. C. Greubel_, Aug 27 2024
%e A143228 First few rows of the triangle:
%e A143228    1;
%e A143228    1,  1;
%e A143228    2,  2,  4;
%e A143228    3,  3,  6,  9;
%e A143228    5,  5, 10, 15, 25;
%e A143228    7,  7, 14, 21, 35,  49;
%e A143228   11, 11, 22, 33, 55,  77, 121;
%e A143228   15, 15, 30, 45, 75, 105, 165, 225;
%e A143228   ...
%e A143228 T(7,4) = 75 = p(7) * p(4) = 15 * 5.
%t A143228 Table[PartitionsP[n]*PartitionsP[k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 27 2024 *)
%o A143228 (Magma)
%o A143228 A143228:= func< n,k | NumberOfPartitions(n)*NumberOfPartitions(k) >;
%o A143228 [A143228(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 27 2024
%o A143228 (SageMath)
%o A143228 def A143215(n,k): return number_of_partitions(n)*number_of_partitions(k)
%o A143228 flatten([[A143215(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 27 2024
%Y A143228 Cf. A000041, A143229 (row sums).
%Y A143228 Main diagonal gives: A001255.
%K A143228 nonn,tabl
%O A143228 0,4
%A A143228 _Gary W. Adamson_, Jul 31 2008