cp's OEIS Frontend

Values appear in the order determined by A004760(n+1)and A062383(n).

The graph of this sequence looks very elegant.

Table A138138 (a shell model) is another sequence illustrating row patterns for numeric partitions.

Here we assume that the starting offset of A126441 is 0, i.e., A126441(0)=1, A126441(1)=2, etc.

Sequences A000041, A002865 and A053445 count numeric partitions. A125106 maps numeric partitions to the natural numbers and has A000120 elements per row. A126441 has A000041 elements per column and is a tabular arrangement of the natural numbers. A140691 is a rearrangement of table A126441. A140692 extracts the cyclic cases and has A002865 elements per column.

The values for A140760 can also be generated by beginning with A140759 and repeatedly multiplying by two as follows:

2

....6.....12......24.....48.....96

..........14..........28........56

..................26........52....

..................30............60

.........................50.......

.........................54.......

.........................62.......

Note that the number of entries in each column is given by A053445.

Here we assume that the starting offset of A126441 is 0, i.e. A126441(0)=1, A126441(1)=2, etc.

From Emeric Deutsch, Sep 06 2017: (Start)

Row sums of the triangle A161919.

a(n) = sum of the viabin numbers of the partitions of n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [3,1,1] of 5. The southeast border of its Ferrers board yields 10011, leading to the viabin number 19. (End)

First in a series of triangular arrays which comprise subsequences of A096443(n).

The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.

Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i, j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.

To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - George Beck, Jan 10 2011

Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link. - Tilman Piesk, Nov 09 2014

Used to calculate number of subspaces of Zp^n where Zp is field of integers mod p.

Consider a square matrix A and call it special if (0) A is an upper triangular matrix, (1) a nonzero column of A has a 1 on the main diagonal and (2) if a row has a 1 on the main diagonal then this is the only nonzero element in that row.

If the diagonal of a special matrix is given (it can only contain 0's and 1's), many of the fields of A are determined by (0), (1) and (2). The number of fields that can be freely chosen while still satisfying (0), (1) and (2) is a(n), where n is the diagonal, read as a binary number with least significant bit at upper left.

a(n) is also the minimum number of adjacent bit swap operations required to pack all the ones of n to the right. - Philippe Beaudoin, Aug 19 2014

From Rakesh Khanna A, Aug 06 2021: (Start)

a(n) is also the area under the curve formed from the binary representation of n where each 0-bit corresponds to an increase of one unit along the x-axis and each 1-bit corresponds to an increase of one unit along the y-axis.

E.g., n = 20 = 10100_2 and the area under the curve shown below is a(n) = 5.

1 0 1 0 0

\ \ \ \ \ |

\ \ \+----+----+

\ \ | |

\+----+ +

| |

----+----+----+----+

(End)

a(n) gives the one-based position of the first nonzero term on the row n-1 of A126441.

Sequence A016116 can be used to identify the extracted subsequence by computing the number of terms to alternately extract and skip. [This comment is from the original submitter. I don't understand it. - Antti Karttunen, Oct 12 2009]