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%I #43 Sep 26 2023 20:26:11
%S 1,1,1,1,2,1,1,1,0,1,3,2,2,1,0,1,1,1,0,1,5,3,3,2,0,2,1,1,0,1,1,0,0,0,
%T 1,7,5,5,3,0,3,2,2,0,2,1,0,0,0,1,1,1,1,0,0,1,11,7,7,5,0,5,3,3,0,3,2,0,
%U 0,0,2,1,1,1,0,0,1,1,0,0,0,0,0,1
%N Tetrahedron P(n,j,k) = T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.
%C This tetrahedron shows a connection between divisors and partitions.
%C Conjecture 1: P(n,j,k) is the number of partitions of n that contain at least m parts of size k, where m = j/k, if k divides j otherwise P(n,j,k) = 0.
%C Conjecture 2: P(n,j,k) is the number of parts that are the m-th part of size k in all partitions of n, where m = j/k, if k divides j otherwise P(n,j,k) = 0.
%C The sum of all elements of slice n is A006128(n).
%C The sum of row j of slice n is A221530(n,j).
%C The sum of column k of slice n is A066633(n,k).
%C See also the tetrahedron of A221649.
%H Paolo Xausa, <a href="/A221650/b221650.txt">Table of n, a(n) for n = 1..11480</a> (rows n = 1..40 of the tetrahedron, flattened)
%F P(n,j,k) = A051731(j,k)*A000041(n-j) = (1/k)*A221649(n,j,k).
%e First six slices of tetrahedron are
%e ---------------------------------------------------
%e n j k: 1 2 3 4 5 6 A221530 A006128
%e ---------------------------------------------------
%e 1 1 1, 1 1
%e ...................................................
%e 2 1 1, 1
%e 2 2 1, 1, 2 3
%e ...................................................
%e 3 1 2, 2
%e 3 2 1, 1, 2
%e 3 3 1, 0, 1, 2 6
%e ...................................................
%e 4 1 3, 3
%e 4 2 2, 2, 4
%e 4 3 1, 0, 1, 2
%e 4 4 1, 1, 0, 1, 3 12
%e ...................................................
%e 5 1 5, 5
%e 5 2 3, 3, 6
%e 5 3 2, 0, 2, 4
%e 5 4 1, 1, 0, 1, 3
%e 5 5 1, 0, 0, 0, 1, 2 20
%e ...................................................
%e 6 1 7, 7
%e 6 2 5, 5, 10
%e 6 3 3, 0, 3, 6
%e 6 4 2, 2, 0, 2, 6
%e 6 5 1, 0, 0, 0, 1, 2
%e 6 6 1, 1, 1, 0, 0, 1 4 35
%e ...................................................
%t A221650row[n_]:=Flatten[Table[If[Divisible[j,k],PartitionsP[n-j],0],{j,n},{k,j}]];Array[A221650row,10] (* _Paolo Xausa_, Sep 26 2023 *)
%Y Cf. A000005, A006128, A027750, A051731, A066633, A127093, A221530, A221649.
%K nonn,tabf
%O 1,5
%A _Omar E. Pol_, Jan 21 2013