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Search: id:A097306
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| A097306 |
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Array of number of partitions of n with odd parts not exceeding 2*m-1 with m in {1,2,..., ceiling(n/2)}. |
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4
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| 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
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OFFSET
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1,4
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COMMENT
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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).
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LINKS
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W. Lang, First 18 rows.
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FORMULA
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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(A097305(n, k), k=1..m), m=1..ceiling(n/2), n>=1.
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EXAMPLE
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[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).
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MAPLE
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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))].
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CROSSREFS
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Row sums: A097307.
Adjacent sequences: A097303 A097304 A097305 this_sequence A097307 A097308 A097309
Sequence in context: A030718 A028334 A083269 this_sequence A102632 A094076 A089611
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 13 2004
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