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Tetrahedron P(n,j,k) = T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.
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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