A097306 Array of number of partitions of n with odd parts not exceeding 2*m-1 with m in {1, 2, ..., ceiling(n/2)}.

1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 4, 1, 3, 4, 5, 1, 3, 5, 6, 1, 4, 6, 7, 8, 1, 4, 7, 9, 10, 1, 4, 8, 10, 11, 12, 1, 5, 9, 12, 14, 15, 1, 5, 10, 14, 16, 17, 18, 1, 5, 11, 16, 19, 21, 22, 1, 6, 13, 19, 23, 25, 26, 27, 1, 6, 14, 21, 26, 29, 31, 32, 1, 6, 15, 24, 30, 34, 36, 37, 38, 1, 7, 17, 27

Offset

1,4

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 partial row sums array of array A097305.

The number of partitions of N=2*n (n >= 1) into even parts not exceeding 2*k, with k from {1,...,n}, is given by the triangle A026820(n,k).

Links

Table of n, a(n) for n=1..85.

Wolfdieter Lang, First 18 rows.

Formula

T(n, m) = number of partitions of n with odd parts only and largest parts <= 2*m-1 for m from {1, 2, ..., ceiling(n/2)}.

T(n, m) = Sum_{k=1..m} A097305(n, k), m = 1..ceiling(n/2), n >= 1.

Example

[1]; [1]; [1,2]; [1,2]; [1,2,3]; [1,3,4]; [1,3,4,5]; [1,3,5,6]; ...
T(8,2)=3 because there are three partitions of 8 with odd parts not exceeding 3, namely (1^8), (1^5,3) and (1^2,3^2).
T(6,2)=3 from the partitions (1^6), (1^3,3) and (3^2).

Maple

Sequence of row numbers for n>=1: [seq(coeff(series(product(1/(1-x^(2*k-1)), k=1..p), x, n+1), x, n), p=1..ceil(n/2))].

Crossrefs

Row sums: A097307.

Sequence in context: A373947 A028334 A083269 * A102632 A094076 A089611

Adjacent sequences: A097303 A097304 A097305 * A097307 A097308 A097309

Keyword

nonn,tabf,easy

Author

Wolfdieter Lang, Aug 13 2004

Status

approved