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%I #51 Aug 04 2021 03:16:06
%S 1,1,1,2,1,1,2,1,3,1,1,1,2,1,3,1,1,2,4,1,2,1,1,1,2,1,3,1,1,2,4,1,2,1,
%T 1,5,1,3,1,2,1,1,1,1,2,1,3,1,1,2,4,1,2,1,1,5,1,3,1,2,1,1,1,2,3,6,1,2,
%U 4,1,3,1,2,1,2,1,1,1,7,1,5,1,2,4,1,3,1,3,1,2,1,2,1,1,1,1
%N Irregular triangle read by rows which is constructed in row n replacing the first A000070(n-1) terms of A336811 with their divisors.
%C The terms in row n are also all parts of all partitions of n.
%C The terms of row n in nonincreasing order give the n-th row of A302246.
%C The terms of row n in nondecreasing order give the n-th row of A302247.
%C For further information about the correspondence divisor/part see A336811 and A338156.
%e Triangle begins:
%e [1];
%e [1],[1, 2];
%e [1],[1, 2],[1, 3],[1];
%e [1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1];
%e [1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1],[1, 5],[1, 3],[1, 2],[1],[1];
%e ...
%e Below the table shows the correspondence divisor/part.
%e |---|-----------------|-----|-------|---------|-----------|-------------|
%e | n | | 1 | 2 | 3 | 4 | 5 |
%e |---|-----------------|-----|-------|---------|-----------|-------------|
%e | P | | | | | | |
%e | A | | | | | | |
%e | R | | | | | | |
%e | T | | | | | | 5 |
%e | I | | | | | | 3 2 |
%e | T | | | | | 4 | 4 1 |
%e | I | | | | | 2 2 | 2 2 1 |
%e | O | | | | 3 | 3 1 | 3 1 1 |
%e | N | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 |
%e | S | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 |
%e ----|-----------------|-----|-------|---------|-----------|-------------|
%e .
%e |---|-----------------|-----|-------|---------|-----------|-------------|
%e | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 |
%e | L | | | | |/| | |/|/| | |/|/|/| | |/|/|/|/| |
%e | I | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |
%e | N | | * | * * | * * * | * * * * | * * * * * |
%e | K | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |
%e | | | = | = = | = = = | = = = = | = = = = = |
%e | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |
%e |---|-----------------|-----|-------|---------|-----------|-------------|
%e .
%e . |-------|
%e . |Section|
%e |---|-------|---------|-----|-------|---------|-----------|-------------|
%e | | 1 | A000012 | 1 | 1 | 1 | 1 | 1 |
%e | |-------|---------|-----|-------|---------|-----------|-------------|
%e | | 2 | A000034 | | 1 2 | 1 2 | 1 2 | 1 2 |
%e | |-------|---------|-----|-------|---------|-----------|-------------|
%e | D | 3 | A010684 | | | 1 3 | 1 3 | 1 3 |
%e | I | | A000012 | | | 1 | 1 | 1 |
%e | V |-------|---------|-----|-------|---------|-----------|-------------|
%e | I | 4 | A069705 | | | | 1 2 4 | 1 2 4 |
%e | S | | A000034 | | | | 1 2 | 1 2 |
%e | O | | A000012 | | | | 1 | 1 |
%e | R |-------|---------|-----|-------|---------|-----------|-------------|
%e | S | 5 | A010686 | | | | | 1 5 |
%e | | | A010684 | | | | | 1 3 |
%e | | | A000034 | | | | | 1 2 |
%e | | | A000012 | | | | | 1 |
%e | | | A000012 | | | | | 1 |
%e |---|-------|---------|-----|-------|---------|-----------|-------------|
%e .
%e In the above table both the zone of partitions and the "Link" zone are the same zones as in the table of the example section of A338156, but here in the lower zone the divisors are ordered in accordance with the sections of the set of partitions of n.
%e The number of rows in the j-th section of the lower zone is equal to A000041(j-1).
%e The divisors of the j-th section are also the parts of the j-th section of the set of partitions of n.
%Y Another version of A338156.
%Y Row n has length A006128(n).
%Y The sum of row n is A066186(n).
%Y The product of row n is A007870(n).
%Y Row n lists the first n rows of A336812.
%Y The number of parts k in row n is A066633(n,k).
%Y The sum of all parts k in row n is A138785(n,k).
%Y The number of parts >= k in row n is A181187(n,k).
%Y The sum of all parts >= k in row n is A206561(n,k).
%Y The number of parts <= k in row n is A210947(n,k).
%Y The sum of all parts <= k in row n is A210948(n,k).
%Y Cf. A000012, A000034, A000041, A000070, A002260, A010684, A010686, A027750, A066633, A069705, A135010, A138785, A181187, A221529, A221649, A237593, A302246, A302247, A336811, A340011, A340031, A340032, A340035, A340056, A340057.
%K nonn,tabf
%O 1,4
%A _Omar E. Pol_, Jul 31 2021
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