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Search: id:A058398
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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
(list; table; graph; listen)
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OFFSET
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1,9
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COMMENT
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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.
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LINKS
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H. Bottomley, Illustration of initial terms
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FORMULA
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a(n, m)= A008284(n, n-(m-1)).
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). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jul 25 2006
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EXAMPLE
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1; 1,1; 1,1,1; 1,1,2,1; 1,1,2,2,1;... (lower triangular matrix)
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CROSSREFS
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Cf. A026820, A008284, A000041.
Sequence in context: A086074 A089723 A055215 this_sequence A091499 A137350 A166240
Adjacent sequences: A058395 A058396 A058397 this_sequence A058399 A058400 A058401
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KEYWORD
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nonn,easy,tabl,nice
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Dec 11 2000
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