cp's OEIS Frontend

In n is odd then row n lists the numbers of row n of A135010. If n is even the row n lists the 1's of row n of A135010 and then row n lists the other numbers of row n of A135010 in reverse order.

Volumes of the parts in the section model of partitions version "boxes" in which each part of size k has a volume = k^2. Row sums of this triangle give A206440 and partial sums of A206440 give A066183.

If n is odd then row n lists the numbers of row n of A141285. If n is even then row n lists the numbers of row n of A141285 but in reverse order.

This sequence shows the growth of the shell model of A135010 step by step. At stage n one part of size A135010(n) is added to the structure.

Consider a physical model of the partitions of n in which each part p of size A135010(n,j) is represented by a right circular cylinder with radius j and height 2. T(n,k) is also the distance (or coordinate X) from the axis Y to the center of the base of cylinder of the part p in the structure of A135010.

Or, in the list of natural numbers (A000027), replace n with its divisors.

This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.

Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008

Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011

{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012

Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012

Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013

The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015

The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017

T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019

Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025

All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

Mirror of triangle A135010.

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009

Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009

Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013

Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016

From Gus Wiseman, May 22 2021: (Start)

The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.

For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:

1 . 3 . 5 . 7 . 9 . B . D

21 41 43 63 65 85

221 61 81 83 A3

421 441 A1 C1

2221 621 443 643

4221 641 661

22221 821 841

4421 A21

6221 4441

42221 6421

222221 8221

44221

62221

422221

2222221

Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:

(11) (22) (33) (44) (55) (66)

(211) (321) (422) (532) (633)

(3111) (431) (541) (642)

(4211) (5221) (651)

(41111) (5311) (6222)

(52111) (6321)

(511111) (6411)

(62211)

(63111)

(621111)

(6111111)

Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:

(2) (22) (222) (2222) (22222) (222222) (2222222)

(31) (321) (3221) (32221) (322221) (3222221)

(411) (3311) (33211) (332211) (3322211)

(4211) (42211) (333111) (3332111)

(5111) (43111) (422211) (4222211)

(52111) (432111) (4322111)

(61111) (441111) (4331111)

(522111) (4421111)

(531111) (5222111)

(621111) (5321111)

(711111) (5411111)

(6221111)

(6311111)

(7211111)

(8111111)

(End)