A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 4, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 5, 2, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2

Offset

1,2

Comments

The remainder of row n is necessarily A000041(n-1) 1's.

Previous name: A modular table of partitions. Row n lists the parts of the last section of the set of partitions of n.

Row n lists the nonzero terms of the n-th row of A138136 together with A000041(n-1) 1's.

Row n is also the n-th row of A138138 in reverse order.

Links

Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.

Omar E. Pol, Illustration of the rows 1..6 of the triangle

Example

Triangle begins:
1
2,1
3,1,1
4,2,2,1,1,1
5,3,2,1,1,1,1,1,
6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

Mathematica

Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1] ~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 8}] // Flatten (* Robert Price, May 22 2020 *)

Crossrefs

Mirror of A138138.

Row lengths give A138137.

Row sums give A138879.

Column 1 gives A000027.

Right border gives A000012.

Another version is A138121 which is the mirror of A135010.

Cf. A000041, A080577, A138136, A139094, A139100.

Sequence in context: A355145 A116599 A138121 * A207378 A166556 A275257

Adjacent sequences: A138148 A138149 A138150 * A138152 A138153 A138154

Keyword

nonn,tabf,changed

Author

Omar E. Pol, Mar 21 2008

Extensions

New name and comments edited by Peter Munn and Omar E. Pol, Jul 25 2025

Status

approved