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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.

%I A060850 #47 Sep 15 2018 02:08:51
%S A060850 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,
%T A060850 27,65,105,108,65,15,1,8,35,98,190,252,221,110,22,1,9,44,140,315,506,
%U A060850 574,429,185,30,1,10,54,192,490,918,1265,1240,810,300,42,1,11,65,255
%N 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.
%C A060850 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
%F A060850 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
%e A060850 Table (row k, k >= 0: number of partitions of n, n >= 0, into parts of k kinds):
%e A060850 Array begins:
%e A060850 =======================================================================
%e A060850 k\n| 0   1   2    3     4     5      6       7       8       9       10
%e A060850 ---|-------------------------------------------------------------------
%e A060850 1  | 1   1   2    3     5     7     11      15      22      30       42
%e A060850 2  | 1   2   5   10    20    36     65     110     185     300      481
%e A060850 3  | 1   3   9   22    51   108    221     429     810    1479     2640
%e A060850 4  | 1   4  14   40   105   252    574    1240    2580    5180    10108
%e A060850 5  | 1   5  20   65   190   506   1265    2990    6765   14725    31027
%e A060850 6  | 1   6  27   98   315   918   2492    6372   15525   36280    81816
%e A060850 7  | 1   7  35  140   490  1547   4522   12405   32305   80465   192899
%e A060850 8  | 1   8  44  192   726  2464   7704   22528   62337  164560   417140
%e A060850 9  | 1   9  54  255  1035  3753  12483   38709  113265  315445   841842
%e A060850 10 | 1  10  65  330  1430  5512  19415   63570  195910  573430  1605340
%e A060850 11 | 1  11  77  418  1925  7854  29183  100529  325193  997150  2919411
%e A060850   ...
%e A060850 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)
%e A060850 Triangle begins: (column k: n - k kinds of parts)
%e A060850 ===================================
%e A060850 n\k| 0   1   2   3   4   5   6   7
%e A060850 ---+-------------------------------
%e A060850 0  |(1)
%e A060850 1  | 1, (0)
%e A060850 2  | 1,  1, (0)
%e A060850 3  | 1,  2,  2, (0)
%e A060850 4  | 1,  3,  5,  3, (0)
%e A060850 5  | 1,  4,  9, 10,  5, (0)
%e A060850 6  | 1,  5, 14, 22, 20,  7, (0)
%e A060850 7  | 1,  6, 20, 40, 51, 36, 11, (0)
%e A060850   ...
%t A060850 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 A060850 t[n_, k_]; = IntegerPartitions[n, {k}]; Table[ t[n - k + 1, k], {n, 12}, {k, n}] // Flatten (* _Robert G. Wilson v_, Aug 08 2018 *)
%Y A060850 Cf. A067687 (table antidiagonal sums, triangle row sums).
%Y A060850 Rows (table), diagonals (triangle): A000041, A000712, A000716, A023003-A023021, A006922.
%Y A060850 Columns (table, triangle): A000012, A001477, A000096, A006503, A006504.
%K A060850 tabl,nonn,easy
%O A060850 1,5
%A A060850 Bo T. Ahlander (ahlboa(AT)isk.kth.se), May 03 2001
%E A060850 More terms from _Vladeta Jovovic_, Jan 02 2004