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A058398 Partition triangle A008284 read from right to left. 5
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; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

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.

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 27.

LINKS

Table of n, a(n) for n=1..98.

H. Bottomley, Illustration of initial terms

FORMULA

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). - Franklin T. Adams-Watters, Jul 25 2006

EXAMPLE

1; 1,1; 1,1,1; 1,1,2,1; 1,1,2,2,1;... (lower triangular matrix)

MATHEMATICA

row[n_] := Table[ IntegerPartitions[n, k] // Length, {k, 0, n}] // Differences // Reverse; Table[row[n], {n, 1, 14}] // Flatten (* Jean-Fran├žois Alcover, Feb 28 2013 *)

CROSSREFS

Cf. A026820, A008284, A000041.

Sequence in context: A276317 A055215 A239550 * A091499 A284249 A137350

Adjacent sequences:  A058395 A058396 A058397 * A058399 A058400 A058401

KEYWORD

nonn,easy,tabl,nice

AUTHOR

Wolfdieter Lang, Dec 11 2000

STATUS

approved

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Last modified April 27 04:48 EDT 2017. Contains 285507 sequences.