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A097306 Array of number of partitions of n with odd parts not exceeding 2*m-1 with m in {1, 2, ..., ceiling(n/2)}. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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: A274225 A028334 A083269 * A102632 A094076 A089611
KEYWORD
nonn,tabf,easy
AUTHOR
Wolfdieter Lang, Aug 13 2004
STATUS
approved

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Last modified April 13 05:19 EDT 2024. Contains 371639 sequences. (Running on oeis4.)