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%I #14 Mar 13 2013 18:06:11
%S 1,3,4,2,7,6,6,17,1,12,29,7,8,55,23,15,84,58,3,13,122,134,13,18,181,
%T 249,52,12,240,464,140,3,28,321,765,348,17,14,407,1249,746,69,24,546,
%U 1875,1501,220,1,24,628,2835,2793,586,13,31,828,4024,4927,1431,56,18,940,5707,8331,3123,215,39,1211,7741,13520,6436,650,4
%N Triangular array read by rows. T(n,k) is the number of partitions of n (using 1 type of part 1, 2 types of part 2, ..., i types of part i, ...) that have exactly k distinct parts.
%C Row sums = A000219.
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; page 171
%F O.g.f.: Product_{i>=1} (1 + y*x^i/(1-x^i))^i.
%e 1,
%e 3,
%e 4, 2,
%e 7, 6,
%e 6, 17, 1,
%e 12, 29, 7,
%e 8, 55, 23,
%e 15, 84, 58, 3,
%e 13, 122, 134, 13,
%e 18, 181, 249, 52,
%e 12, 240, 464, 140, 3,
%e 28, 321, 765, 348, 17,
%e 14, 407, 1249, 746, 69,
%e 24, 546, 1875, 1501, 220, 1
%e 24, 628, 2835, 2793, 586, 13
%e T(4,2) = 6 because we have: 3+1, 3'+1, 3''+1, 2+2', 2+1+1, 2'+1+1.
%t nn=15;f[list_]:=Select[list,#>0&];Map[f,Drop[CoefficientList[Series[ Product[(1+y x^i/(1-x^i))^i,{i,1,nn}],{x,0,nn}],{x,y}],1]]//Grid
%K nonn,tabf
%O 1,2
%A _Geoffrey Critzer_, Mar 13 2013
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