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%I #48 Sep 27 2023 02:42:21
%S 1,1,1,2,2,1,2,1,0,3,3,2,4,1,0,3,1,2,0,4,5,3,6,2,0,6,1,2,0,4,1,0,0,0,
%T 5,7,5,10,3,0,9,2,4,0,8,1,0,0,0,5,1,2,3,0,0,6,11,7,14,5,0,15,3,6,0,12,
%U 2,0,0,0,10,1,2,3,0,0,6,1,0,0,0,0,0,7
%N Tetrahedron E(n,j,k) = k*T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.
%C The tetrahedron shows a connection between divisors and partitions.
%C The sum of all elements of slice n is A066186(n).
%C The sum of row j of slice n is A221529(n,j).
%C The sum of column k of slice n is A138785(n,k), the sum of all parts of size k in all partitions of n.
%C See also the tetrahedron of A221650.
%H Paolo Xausa, <a href="/A221649/b221649.txt">Table of n, a(n) for n = 1..11480</a> (rows n = 1..40 of the tetrahedron, flattened)
%F E(n,j,k) = k*A051731(j,k)*A000041(n-j) = A127093(j,k)*A000041(n-j) = k*A221650(n,j,k).
%e First five slices of tetrahedron are
%e ---------------------------------------------------
%e n j / k 1 2 3 4 5 6 A221529 A066186
%e ---------------------------------------------------
%e 1 1 1, 1 1
%e ...................................................
%e 2 1 1, 1
%e 2 2 1, 2, 3 4
%e ...................................................
%e 3 1 2, 2
%e 3 2 1, 2, 3
%e 3 3 1, 0, 3, 4 9
%e ...................................................
%e 4 1 3, 3
%e 4 2 2, 4, 6
%e 4 3 1, 0, 3, 4
%e 4 4 1, 2, 0, 4, 7 20
%e ...................................................
%e 5 1 5, 5
%e 5 2 3, 6, 9
%e 5, 3, 2, 0, 6, 8
%e 5, 4, 1, 2, 0, 4, 7
%e 5, 5, 1, 0, 0, 0, 5, 6 35
%e ...................................................
%e .
%e From _Omar E. Pol_, Jul 26 2021: (Start)
%e The slices of the tetrahedron appear in the upper zone of the following table (formed by four zones) which shows the correspondence between divisors and parts (n = 1..5):
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | n | | 1 | 2 | 3 | 4 | 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | - | | | | | 5 |
%e | C | - | | | | 3 | 3 6 |
%e | O | - | | | 2 | 2 4 | 2 0 6 |
%e | N | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 |
%e | D | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | D | A127093 | | | | | 1 |
%e | I |---------|-----|-------|---------|-----------|-------------|
%e | V | A127093 | | | | 1 | 1 2 |
%e | I | A127093 | | | | 1 | 1 2 |
%e | S | A127093 | | | | 1 | 1 2 |
%e | O |---------|-----|-------|---------|-----------|-------------|
%e | R | A127093 | | | 1 | 1 2 | 1 0 3 |
%e | S | A127093 | | | 1 | 1 2 | 1 0 3 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |
%e | | | = | = = | = = = | = = = = | = = = = = |
%e | L | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |
%e | I | | * | * * | * * * | * * * * | * * * * * |
%e | N | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |
%e | K | | | | |\| | |\|\| | |\|\|\| | |\|\|\|\| |
%e | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | P | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 |
%e | A | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 |
%e | R | | | | 3 | 3 1 | 3 1 1 |
%e | T | | | | | 2 2 | 2 2 1 |
%e | I | | | | | 4 | 4 1 |
%e | T | | | | | | 3 2 |
%e | I | | | | | | 5 |
%e | O | | | | | | |
%e | N | | | | | | |
%e | S | | | | | | |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e The upper zone is a condensed version of the "divisors" zone.
%e The above table is the table of A340011 upside down.
%e For more information about the correspondence divisor/part see A338156. (End)
%t A221649row[n_]:=Flatten[Table[If[Divisible[j,k],PartitionsP[n-j]k,0],{j,n},{k,j}]];Array[A221649row,10] (* _Paolo Xausa_, Sep 26 2023 *)
%Y Nonzero terms give A340057.
%Y Cf. A000005, A000041, A000203, A027750, A051731, A066186, A127093, A138785, A221529, A221650, A237593, A336811, A336812, A338156, A340011, A340031, A340032, A340035, A340056.
%K nonn,tabf
%O 1,4
%A _Omar E. Pol_, Jan 21 2013
%E a(18)-a(19) and a(28)-a(29) corrected by _Paolo Xausa_, Sep 26 2023
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