OFFSET
1,5
COMMENTS
The square is the following table:
1 1 1 1 1 1 1...
1 2 2 2 2 2 2...
1 2 3 3 3 3 3...
1 3 4 5 5 5 5...
1 3 5 6 7 7 7...
1 4 7 9 10 11 11...
1 4 8 11 13 14 15...
REFERENCES
Ivan Niven, "Mathematics of Choice, How to Count Without Counting", MAA, 1965, pp. 98-99 (table p. 98).
LINKS
Martin Y. Champel, Table of n, a(n) for n = 1..405
A. Comtet, S. N. Majumdar and S. Ouvry, Integer partitions and exclusion statistics, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 11255-11269.
A. Comtet et al., Integer Partitions and Exclusion Statistics, arXiv:0705.2640 [cond-mat.stat-mech], 2007, eq. (12).
FORMULA
Antidiagonals of table of values of p_k(n) (the number of partitions of n with no summand greater than k).
T(n,m) = sum_{i=1..m} A008284(n,i). T(n,m) = A026820(n,m) if m<=n and T(n,m)=T(n,n) if m>=n. G.f. column m: 1/(1-x)/(1-x^2)/.../(1-x^m) = sum_(n=1,2,3..) T(n,m) x^n [Comtet]. - R. J. Mathar, Aug 31 2007
EXAMPLE
The partitions of 6 are:
1 + 1 + 1 + 1 + 1 + 1; 2 + 1 + 1 + 1 + 1; 3 + 1 + 1 + 1; 2 + 2 + 1 + 1; 4 + 1 + 1; 3 + 2 + 1; 2 + 2 + 2; 5 + 1, 4 + 2, 3 + 3, 6.
There are 9 partitions of 6 having summands no larger than 4, so p_4(6) = 9.
MATHEMATICA
T[n_, k_] := T[n, k] = If[n == 0 || k == 1, 1, T[n, k-1] + If[k > n, 0, T[n-k, k]]];
Table[T[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 08 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Apr 27 2005
EXTENSIONS
Edited, corrected and extended by Franklin T. Adams-Watters, Jan 12 2006
STATUS
approved