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A058398
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Partition triangle A008284 read from right to left.
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89
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 5, 5, 4, 1, 1, 1, 2, 3, 5, 6, 7, 4, 1, 1, 1, 2, 3, 5, 7, 9, 8, 5, 1, 1, 1, 2, 3, 5, 7, 10, 11, 10, 5, 1, 1, 1, 2, 3, 5, 7, 11, 13, 15, 12, 6, 1, 1, 1, 2, 3, 5, 7, 11, 14, 18, 18, 14, 6, 1, 1, 1, 2, 3, 5, 7, 11
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OFFSET
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1,9
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COMMENTS
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a(n,m) is the number of partitions of n with n-(m-1) parts or, equivalently, with greatest part n-(m-1).
The columns are the diagonals of triangle A008284. The diagonals are the columns of the partition array p(n,m), n >= 0, m >= 1, with p(n,m) the number of partitions of n in which every part is <= m; p(0,m) := 1. For n >= 1 this array is obtained from table A026820 read as lower triangular array with extension of the rows according to p(n,m)=A000041(n) for m>n.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 27.
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LINKS
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FORMULA
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a(n, m)= p(m-1, n-m+1), n >= m >= 1 with the p(n, m) array defined in the comment.
a(n, m)=0 if n<m or m<=0 or n=0; a(1, 1)=1; a(n, m)= a(n-1, m)+a(m-1, 2*m-n+1).
Viewed as a square array by antidiagonals, T(n,k) = 0 if n<0; T(n,1) = 1; otherwise T(n,k) = T(n,k-1) + T(n-k,k). - Franklin T. Adams-Watters, Jul 25 2006
Let x be a triangular number C(n,2), where n is the integer being partitioned. Then a(x) = a(x+1) = a(x+2) = 1. Also, a(x+3) = 2 for x>3 and a(x-1) = floor(n/2). - Allan Bickle, Apr 18 2024
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EXAMPLE
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Lower triangular matrix:
1;
1,1;
1,1,1;
1,1,2,1;
1,1,2,2,1;
1,1,2,3,3,1;
1,1,2,3,4,3,1;
1,1,2,3,5,5,4,1;
...
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MATHEMATICA
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row[n_] := Table[ IntegerPartitions[n, k] // Length, {k, 0, n}] // Differences // Reverse; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 28 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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