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A106254 Partition table in square format. 1
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 4, 5, 5, 3, 2, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Aug 08 2018, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A285521 A187451 A134542 * A117147 A111007 A176353

Adjacent sequences:  A106251 A106252 A106253 * A106255 A106256 A106257

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

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Last modified April 22 11:08 EDT 2021. Contains 343174 sequences. (Running on oeis4.)