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