A216221 - OEIS

%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