A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.
1, 1, 2, 1, 1, 0, 3, 1, 2, 2, 1, 2, 0, 4, 1, 0, 3, 2, 4, 3, 1, 0, 0, 0, 5, 1, 2, 0, 4, 2, 0, 6, 3, 6, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 2, 4, 0, 8, 3, 0, 9, 5, 10, 7, 1, 0, 0, 0, 0, 0, 7, 1, 2, 3, 0, 0, 6, 2, 0, 0, 0, 10, 3, 6, 0, 12, 5, 0, 15, 7, 14, 11, 1, 2, 0, 4, 0, 0, 0, 8
Offset: 1
Examples
Triangle begins: [1]; [1, 2], [1]; [1, 0, 3], [1, 2], [2]; [1, 2, 0, 4], [1, 0, 3], [2, 4], [3]; [1, 0, 0, 0, 5], [1, 2, 0, 4], [2, 0, 6], [3, 6], [5]; [... Row sums give A066186. Written as an irregular tetrahedron the first five slices are: -- 1; ----- 1, 2, 1; -------- 1, 0, 3, 1, 2, 2; ----------- 1, 2, 0, 4, 1, 0, 3, 2, 4, 3; -------------- 1, 0, 0, 0, 5, 1, 2, 0, 4, 2, 0, 6, 3, 6, 5; -------------- Row sums give A339106. The following table formed by four zones shows the correspondence between divisor and parts (n = 1..5): . |---|---------|-----|-------|---------|-----------|-------------| | n | | 1 | 2 | 3 | 4 | 5 | |---|---------|-----|-------|---------|-----------|-------------| | P | | | | | | | | A | | | | | | | | R | | | | | | | | T | | | | | | 5 | | I | | | | | | 3 2 | | T | | | | | 4 | 4 1 | | I | | | | | 2 2 | 2 2 1 | | O | | | | 3 | 3 1 | 3 1 1 | | N | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 | | S | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 | |---|---------|-----|-------|---------|-----------|-------------| . |---|---------|-----|-------|---------|-----------|-------------| | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 | | L | | | | |/| | |/|/| | |/|/|/| | |/|/|/|/| | | I | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 | | N | | * | * * | * * * | * * * * | * * * * * | | K | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 | | | | = | = = | = = = | = = = = | = = = = = | | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 | |---|---------|-----|-------|---------|-----------|-------------| . |---|---------|-----|-------|---------|-----------|-------------| | | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 | | |---------|-----|-------|---------|-----------|-------------| | | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 | | |---------|-----|-------|---------|-----------|-------------| | D | A127093 | | | 1 | 1 2 | 1 0 3 | | I | A127093 | | | 1 | 1 2 | 1 0 3 | | V |---------|-----|-------|---------|-----------|-------------| | I | A127093 | | | | 1 | 1 2 | | S | A127093 | | | | 1 | 1 2 | | O | A127093 | | | | 1 | 1 2 | | R |---------|-----|-------|---------|-----------|-------------| | S | A127093 | | | | | 1 | | | A127093 | | | | | 1 | | | A127093 | | | | | 1 | | | A127093 | | | | | 1 | | | A127093 | | | | | 1 | |---|---------|-----|-------|---------|-----------|-------------| . |---|---------|-----|-------|---------|-----------|-------------| | | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 | | C | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 | | O | - | | | 2 | 2 4 | 2 0 6 | | N | - | | | | 3 | 3 6 | | D | - | | | | | 5 | |---|---------|-----|-------|---------|-----------|-------------| . This lower zone of the table is a condensed version of the "divisors" zone.
Links
- Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows 1..40 of the triangle, flattened)
Crossrefs
Row sums give A066186.
Nonzero terms give A340056.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340032, A340035, A340061.
Programs
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Mathematica
A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}]; A340011row[n_]:=Flatten[Table[A127093row[n-m+1]PartitionsP[m-1],{m,n}]]; Array[A340011row,10] (* Paolo Xausa, Sep 28 2023 *)
Comments