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Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.
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%I #17 May 09 2018 09:58:54

%S 1,1,1,3,1,4,10,60,1,5,15,140,280,1260,12600,1,6,21,224,630,3780,

%T 34650,110880,360360,2522520,37837800,1,7,28,336,1050,7392,74844,

%U 276276,1513512,9459450,131171040,428828400,2058376320,9777287520,97772875200,2053230379200

%N Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.

%H Alois P. Heinz, <a href="/A262078/b262078.txt">Columns k = 0..36, flattened</a>

%e Triangle T(n,k) begins:

%e : 1;

%e : 1;

%e : 1;

%e : 3, 1;

%e : 4, 1;

%e : 10, 5, 1;

%e : 60, 15, 6, 1;

%e : 140, 21, 7, 1;

%e : 280, 224, 28, 8, 1;

%e : 1260, 630, 336, 36, 9, 1;

%e : 12600, 3780, 1050, 480, 45, 10, 1;

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,`if`(n=0, 1,

%p b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))

%p end:

%p T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):

%p seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);

%t b[n_, i_] := b[n, i] = If[i*(i+1)/2<n, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, Binomial[n, i]*b[n-i, i-1]]]]; T[n_, k_] := b[n, k] - If[k==0, 0, b[n, k-1]]; Table[T[n, k], {k, 0, 7}, {n, k, k*(k+1)/2}] // Flatten (* _Jean-François Alcover_, Dec 18 2016, after _Alois P. Heinz_ *)

%Y Row sums give A007837.

%Y Column sums give A262073.

%Y Cf. A000217, A002024, A262071, A262072 (same read by rows).

%K nonn,tabf

%O 0,4

%A _Alois P. Heinz_, Sep 10 2015