1,5

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...

Ivan Niven, "Mathematics of Choice, How to Count Without Counting", MAA, 1965, pp. 98-99 (table p. 98).

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:cond-mat.stat-mech/0705.2640, eq. (12).

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 (mathar(AT)strw.leidenuniv.nl), Aug 31 2007

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.

Essentially the same as A008284, except for missing one diagonal thereof (which would be zero row of this array).

Sequence in context: A115514 A122632 A134542 this_sequence A117147 A111007 A103691

Adjacent sequences: A106251 A106252 A106253 this_sequence A106255 A106256 A106257

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 27 2005

Edited, corrected and extended by Frank Adams-Watters (FrankTAW(AT)netscape.net), Jan 12 2006