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Irregular triangular array read by rows. T(n,k) is the number of n-permutations with exactly k distinct cycle lengths; n>=1, 1<=k<=floor( (-1+(1+8n)^(1/2))/2 ).
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%I #11 May 14 2016 17:22:14

%S 1,1,2,3,6,8,24,50,120,234,120,720,1764,630,5040,11808,7392,40320,

%T 109584,69552,362880,954000,763200,151200,3628800,10628640,8165520,

%U 1330560,39916800,113891040,109010880,25280640,479001600,1486442880,1345687200,381775680,6227020800,18913184640,19773804960,6763236480,87178291200,283465647360,291950568000,102508005600,10897286400

%N Irregular triangular array read by rows. T(n,k) is the number of n-permutations with exactly k distinct cycle lengths; n>=1, 1<=k<=floor( (-1+(1+8n)^(1/2))/2 ).

%C Row sums = A007838.

%H Alois P. Heinz, <a href="/A224211/b224211.txt">Rows n = 1..300, flattened</a>

%F E.g.f.: Product_{i>=1} (1 + y*x)^i/i.

%e : 1;

%e : 1;

%e : 2, 3;

%e : 6, 8;

%e : 24, 50;

%e : 120, 234, 120;

%e : 720, 1764, 630;

%e : 5040, 11808, 7392;

%e : 40320, 109584, 69552;

%e : 362880, 954000, 763200, 151200;

%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,

%p `if`(i<1, 0, b(n, i-1)+ (i-1)!*b(n-i, i-1)*

%p `if`(i>n, 0, binomial(n, i)*x))))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):

%p seq(T(n), n=1..15); # _Alois P. Heinz_, Oct 21 2015

%t nn=15;f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Product[(1+y x^i/i),{i,1,nn}],{x,0,nn}],{x,y}],1]]//Grid

%K nonn,tabf

%O 1,3

%A _Geoffrey Critzer_, Apr 01 2013