A060850 Array of the coefficients A(n,k) in the expansion of Product_{i>=1} 1/(1-x^i)^n = Sum_{k>=0} A(n,k)*x^k, n >= 1, k >= 0.

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 7, 1, 6, 20, 40, 51, 36, 11, 1, 7, 27, 65, 105, 108, 65, 15, 1, 8, 35, 98, 190, 252, 221, 110, 22, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 1, 11, 65, 255

Offset

1,5

Comments

Table read by antidiagonals: entry (n,k) gives number of partitions of n objects into parts of k kinds. - Franklin T. Adams-Watters, Dec 28 2006

Links

Table of n, a(n) for n=1..70.

Formula

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=1..n} A000041(k-1)*A(n-k;x)*x^(k-1), A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004

Example

Table (row k, k >= 0: number of partitions of n, n >= 0, into parts of k kinds):
Array begins:
=======================================================================
k\n| 0   1   2    3     4     5      6       7       8       9       10
---|-------------------------------------------------------------------
1  | 1   1   2    3     5     7     11      15      22      30       42
2  | 1   2   5   10    20    36     65     110     185     300      481
3  | 1   3   9   22    51   108    221     429     810    1479     2640
4  | 1   4  14   40   105   252    574    1240    2580    5180    10108
5  | 1   5  20   65   190   506   1265    2990    6765   14725    31027
6  | 1   6  27   98   315   918   2492    6372   15525   36280    81816
7  | 1   7  35  140   490  1547   4522   12405   32305   80465   192899
8  | 1   8  44  192   726  2464   7704   22528   62337  164560   417140
9  | 1   9  54  255  1035  3753  12483   38709  113265  315445   841842
10 | 1  10  65  330  1430  5512  19415   63570  195910  573430  1605340
11 | 1  11  77  418  1925  7854  29183  100529  325193  997150  2919411
  ...
Triangle (row n, n >= 0: number of partitions of n into parts of n - k kinds, 0 <= k <= n) (antidiagonals of above table) (parenthesized last term on each row, which would correspond to row k = 0 in above table)
Triangle begins: (column k: n - k kinds of parts)
===================================
n\k| 0   1   2   3   4   5   6   7
---+-------------------------------
0  |(1)
1  | 1, (0)
2  | 1,  1, (0)
3  | 1,  2,  2, (0)
4  | 1,  3,  5,  3, (0)
5  | 1,  4,  9, 10,  5, (0)
6  | 1,  5, 14, 22, 20,  7, (0)
7  | 1,  6, 20, 40, 51, 36, 11, (0)
  ...

Mathematica

t[n_, k_] := CoefficientList[ Series[ Product[1/(1 - x^i)^n, {i, k}], {x, 0, k}], x][[k]]; (* Robert G. Wilson v, Aug 08 2018 *)
t[n_, k_]; = IntegerPartitions[n, {k}]; Table[ t[n - k + 1, k], {n, 12}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 08 2018 *)

Crossrefs

Cf. A067687 (table antidiagonal sums, triangle row sums).

Rows (table), diagonals (triangle): A000041, A000712, A000716, A023003-A023021, A006922.

Columns (table, triangle): A000012, A001477, A000096, A006503, A006504.

Sequence in context: A086350 A239830 A140767 * A208336 A038137 A073133

Adjacent sequences: A060847 A060848 A060849 * A060851 A060852 A060853

Keyword

tabl,nonn,easy

Author

Bo T. Ahlander (ahlboa(AT)isk.kth.se), May 03 2001

Extensions

More terms from Vladeta Jovovic, Jan 02 2004

Status

approved