A097305 Array of number of partitions of n with odd parts only and largest part 2*m-1 with m in {1,2,..., ceiling(n/2)}.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 3, 3, 2, 1, 1, 3, 4, 2, 1, 1, 1, 4, 4, 3, 2, 1, 1, 4, 5, 4, 2, 1, 1, 1, 4, 6, 5, 3, 2, 1, 1, 5, 7, 6, 4, 2, 1, 1, 1, 5, 8, 7, 5, 3, 2, 1, 1, 5, 9, 9, 6, 4, 2, 1, 1, 1, 6, 10, 10, 8, 5, 3, 2, 1, 1, 6, 11, 12, 10, 6, 4, 2, 1, 1, 1, 6

Offset

1,11

Comments

The sequence of row lengths of this array is A008619 = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,...].

This is the first difference array of A097306.

The number of partitions of N=2*n (n>=1) into even parts with largest part 2*k, with k from 1,..,n, is given by the triangle A008284(n,k).

Links

Table of n, a(n) for n=1..102.

W. Lang, First 18 rows.

Formula

T(n, m)= number of partitions of n with only odd parts and largest part is k:=2*m-1, m=1, 2, ..., ceiling(n/2).

Example

[1]; [1]; [1,1]; [1,1]; [1,1,1]; [1,2,1]; [1,2,1,1]; [1,2,2,1]; ...
T(8,2)=2 because there are two partitions of 8 with odd parts from {1,3} and 3 appears at least once, namely (1^5,3) and (1^2,3^2).
T(6,2)=2 from 6= 3+3 = 1+1+1+3.

Crossrefs

Row sums: A000009.

Sequence in context: A355242 A191774 A262885 * A120675 A072699 A143589

Adjacent sequences: A097302 A097303 A097304 * A097306 A097307 A097308

Keyword

nonn,tabf,easy

Author

Wolfdieter Lang, Aug 13 2004

Status

approved